A Proportional-Arithmetic Framework for Fourier Analysis on the Positive Real Line

Abstract

This paper develops a Fourier framework internal to proportional arithmetic on the positive real line. We construct the corresponding complex scalar field, differential and integral operators, oscillatory kernel, Fourier transform, and proportional function spaces. A correspondence theorem proves that the representative of the proportional transform is the classical Fourier transform under the logarithmic identification. Consequently, inversion, Plancherel, convolution, Schwartz invariance, and Sobolev characterizations follow by transport. We establish the exact relation with Fourier analysis on the multiplicative group and with the Mellin transform on the imaginary axis. Model resolvent and heat equations illustrate the operational calculus, while a scale-localized profile shows how spectral modulus and phase encode log-scale width and preferred scale. The construction is therefore a systematic proportional-arithmetic realization of classical harmonic analysis, rather than an analytically independent Fourier theory.

1. Introduction

Classical analysis is normally developed over a fixed arithmetic background: addition, multiplication, subtraction and division are those of the real or complex field, and the corresponding notions of derivative, integral, convolution and transform are built on that structure. Non-Diophantine arithmetic starts from a different premise: an arithmetic may be induced on a set by transporting the usual operations through a bijection. Once such an arithmetic is fixed, it is natural to ask which calculus, function spaces and integral transforms are intrinsic to it. This point of view underlies non-Newtonian calculus and related generalized calculi [1,2,3,4,5,6,7,8]. In this perspective, arithmetic is not only a computational convention; it is part of the mathematical structure from which the associated calculus, integration theory and transform theory are induced.

The arithmetic considered in this paper is proportional arithmetic on the positive real line ${\mathbb{R}}_{+}=(0,\infty )$. It is generated by the logarithmic bijection and has the proportional operations

$$x\oplus y=xy,x\ominus y={\displaystyle \frac{x}{y}},x\odot y=x^{\ln y},x\oslash y=x^{1/\ln y}.$$

The neutral element for proportional addition ⊕ is 1, while the neutral element for proportional multiplication ⊙ is e. Thus, 1 is the proportional zero and e is the proportional unit. These operations are not used here as decorative replacements for classical symbols. They define the algebraic language in which positive quantities, ratios and multiplicative variations are represented. Proportional and multiplicative calculi have already appeared in the study of growth models, differential equations and related applications [9,10,11,12,13]. The present paper develops the corresponding Fourier-analysis layer. This formulation may be conceptually useful when the original quantities are naturally positive and when ratios, proportional increments, and multiplicative interactions are part of the model rather than a secondary interpretation imposed after a logarithmic change of variables.

The central question addressed here is the following: once proportional arithmetic is taken as the underlying arithmetic, can one define a Fourier transform, operational calculus, Schwartz class and Sobolev scale that are expressed internally in proportional notation and are mathematically controlled by precise representative theorems? The answer is affirmative. We construct a proportional complex field, a proportional exponential kernel, a proportional Fourier transform, and proportional function spaces. The logarithmic representative is used throughout as an audit and proof device, but the definitions themselves are stated in the proportional algebra. This distinction is important: the paper does not claim that the analytic information is unrelated to classical Fourier analysis. Rather, it shows how the classical Fourier mechanism is transported into a non-Diophantine arithmetic without abandoning the proportional operations in which the model is formulated.

With the symmetric normalization, the classical Fourier transform is one of the standard tools of harmonic analysis and partial differential equations [14,15,16]:

$$\mathcal{F}\left\{f\right\}\left(\omega \right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{\infty }f\left(u\right)e^{-iu\omega }du.$$

In proportional arithmetic the corresponding transform is defined by

$${\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)={\eta }_p\odot {\int }_0^{\infty }{f}_p\left(t\right)\odot {\mathrm{Exp}}_p(\ominus i_p\odot \xi \odot t)d_{\odot }t,\xi \in {\mathbb{R}}_{+},$$

where $i_p$ is the proportional imaginary unit, ${\eta }_p=\mathcal{E}(1/\sqrt{2\pi })$, ${\mathrm{Exp}}_p$ is the proportional exponential and $d_{\odot }t$ is proportional integration. All factors in this formula are proportional objects: proportional multiplication replaces ordinary multiplication, the proportional integral replaces the usual integral, and the proportional exponential kernel replaces the ordinary oscillatory kernel.

The main structural theorem states that, if $f_p$ has representative $f_c$, then

$${\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)=\mathcal{E}\left(\mathcal{F}\left\{f_c\right\}\left(\ln \xi \right)\right).$$

This identity is not used as the definition of ${\mathcal{F}}_p$. Instead, it verifies that the internal proportional construction is compatible with the representative induced by the underlying arithmetic. It also provides the rigorous mechanism for proving inversion, Riemann–Lebesgue, Plancherel, convolution and regularity results. When $\mathcal{F}\left\{f_c\right\}\left(\ln \xi \right)$ happens to be real, the value may be identified with the positive real number $\exp \left(\mathcal{F}\left\{f_c\right\}\left(\ln \xi \right)\right)$; in general, the transform is genuinely ${\mathbb{C}}_p$-valued.

The relation with Mellin-type analysis and Fourier analysis on multiplicative structures is explicitly acknowledged. The measure $dt/t$ and the representative map place the construction near harmonic analysis on the multiplicative group $(0,\infty )$, and this connection is analytically useful [17,18,19,20,21]. Recent work on half-line transforms also emphasizes the interpretation of the Mellin transform as Fourier analysis on the locally compact abelian multiplicative half-line [21]. Related Fourier-type transforms have also been studied in other non-Diophantine settings, including Cantor-type arithmetic frameworks [22]. These precedents make the novelty question delicate and require a precise positioning of the present work. The proportional Fourier transform studied here is not presented as a new classical harmonic-analysis transform and is not the ordinary Mellin transform of $f_p$. It is the Fourier transform of the proportional representative, expressed back inside proportional arithmetic. The contribution is therefore a proportional-arithmetic realization of harmonic analysis on ${\mathbb{R}}_{+}$: the arithmetic is fixed first, then the calculus, integration, complex structure, transform and spaces are built in that arithmetic.

To make this positioning explicit,

Table 1. Comparison of the classical Fourier transform on $u,\omega \in \mathbb{R}$, the group Fourier transform on $t>0$ with Haar measure $dt/t$, the Mellin transform where defined, and the proportional transform on $t,\xi >0$. The final column records the representative relation under $u=\ln t$ and $\omega =\ln \xi$.

Viewpoint	Group Law	Measure	Kernel/Character	Transformed Object and Relation
Classical Fourier	addition on $\mathbb{R}$	$du$	$e^{-iu\omega }$	$f_c\left(u\right)$; the proof bridge under $u=\ln t$.
Group Fourier on ${\mathbb{R}}_{+}$	multiplication	$dt/t$	$t^{-i\omega }$	$g\left(t\right)={\mathsf{\ell }}_p\left(f_p\left(t\right)\right)$; exactly the representative of ${\mathcal{F}}_p$.
Mellin	multiplication	$dt$, equivalently $dt/t$ after absorbing one power of t	$t^{s-1}$, or $t^s$ relative to $dt/t$	An ordinary function $g\left(t\right)$; on $s=-i\omega$, it agrees with the group transform up to normalization.
This paper	$\oplus ,\odot$ on proportional scalars	$d_{\odot }t$	$K_p(t,\xi )$	$f_p\left(t\right)\in {\mathbb{C}}_p$; its representative is the preceding group transform at $\omega =\ln \xi$.

compares the present framework with the nearest classical and non-Diophantine viewpoints. The table is not meant to hide the representative equivalence; rather, it identifies where the proportional formulation adds its own algebraic language. The comparison is made at the level of formulation and mathematical bookkeeping: the analytic bridge is classical, but the objects being defined and manipulated remain proportional.

The Cantor-type constructions in non-Diophantine arithmetic show that Fourier-type transforms can be built after transporting arithmetic through a generator. The present work differs in three specific aspects: the base arithmetic is the proportional field on ${\mathbb{R}}_{+}$; the complex field ${\mathbb{C}}_p$ is introduced as a formal proportional copy of $\mathbb{C}$ compatible with the embedded real proportional field; and the analysis is developed through proportional $L^q$, Schwartz and Sobolev spaces, including Plancherel theory and proportional differential-equation examples. Thus, the contribution is not the transport mechanism itself, but a systematic proportional realization of Fourier analysis on the positive real line, developed while keeping the proportional operations visible throughout; see [22].

The distinction from the ordinary Mellin transform can be made formulaically. The Mellin transform acts directly on a function on ${\mathbb{R}}_{+}$ through multiplicative characters and the Haar measure $dt/t$. By contrast, ${\mathcal{F}}_p$ is defined with proportional multiplication, proportional integration and the proportional kernel; only after applying representatives does one obtain a Fourier integral in the variable $u=\ln t$. Thus, the analytic bridge is classical, while the scalar field, convolution, derivative, kernel and function spaces remain proportional in the original formulation.

Proposition 1 

(Scope of the proportional realization). Let X and Y be classical function spaces and let $X_p={\mathcal{R}}_p^{-1}\left(X\right)$, $Y_p={\mathcal{R}}_p^{-1}\left(Y\right)$. Every operator identity obtained in this paper by conjugation has the form

$$T_p={\mathcal{R}}_p^{-1}T{\mathcal{R}}_p:X_p\to Y_p.$$

Accordingly, such an identity does not constitute an analytically independent theorem beyond the corresponding classical statement. The contribution established here is the coherent internal realization of the scalar field, topology, calculus, kernel, transform, convolution, and function spaces in proportional arithmetic, together with exact group-Fourier/Mellin identification and proportional-variable interpretations.

The main contributions of this paper are as follows:

• We formulate the algebraic foundations of proportional arithmetic on ${\mathbb{R}}_{+}$ as a non-Diophantine field and keep the proportional operations as the primary notation throughout the construction.
• We define proportional differentiation and proportional integration, and we clarify the role of representatives without allowing the representative notation to replace the proportional algebra.
• We introduce proportional complex scalars, a proportional imaginary unit and a proportional exponential kernel, avoiding branch ambiguities of the complex logarithm by working with a formal proportional copy of $\mathbb{C}$.
• We define the proportional Fourier transform on complex-valued proportional $L^1$ spaces and prove the correspondence theorem with the classical Fourier transform of the representative.
• We establish Riemann–Lebesgue, inversion, Plancherel and unitary $L^2$-extension results for the proportional Fourier transform.
• We derive proportional analogues of translation, modulation, scaling, differentiation and convolution identities.
• We construct complex-valued proportional Schwartz and Sobolev spaces, prove invariance of the proportional Schwartz space under ${\mathcal{F}}_p$, and characterize Sobolev regularity in the proportional frequency domain.
• We illustrate the framework through proportional Gaussian-type functions, a genuinely complex transform pair, a proportional resolvent equation and a proportional heat equation.

The novelty claimed here is therefore deliberately specific. We do not claim the existence of a generator-based Fourier transform in abstract non-Diophantine arithmetic as a new idea by itself. The contribution is a systematic proportional realization of the core Fourier package on ${\mathbb{R}}_{+}$, including the proportional complex field, proportional kernel, proportional function spaces, operational calculus, weak Sobolev formulation and model differential equations, all written in the arithmetic in which the original positive-scale variables live.

The paper is organized as follows. Section 2 introduces proportional arithmetic as a non-Diophantine arithmetic. Section 3 develops proportional functions, differentiation and integration. Section 4 introduces proportional complex numbers, complex-valued proportional function spaces and the proportional exponential kernel. Section 5 defines the proportional Fourier transform and proves the correspondence, inversion, Riemann–Lebesgue and Plancherel results. Section 6 establishes the main operational properties. Section 7 and Section 8 introduce proportional Schwartz and Sobolev spaces. Section 9 presents applications to proportional differential equations. Section 10 discusses the relationship with classical Fourier analysis and Mellin-type structures, and Section 11 summarizes the main conclusions.

2. Proportional Arithmetic as a Non-Diophantine Arithmetic

In this section we introduce the algebraic structure underlying proportional calculus. The point of departure is the idea, central in non-Diophantine arithmetic, that the usual arithmetic operations need not be fixed a priori. Instead, new operations may be induced on a given set by transporting the classical operations through a suitable bijection. This principle provides a systematic mechanism for constructing alternative arithmetics and the corresponding differential, integral and functional structures [2,6,7].

Let ${\mathbb{R}}_{+}=(0,\infty )$ and consider the bijection

$$\varphi :{\mathbb{R}}_{+}\to \mathbb{R},\varphi \left(x\right)=\ln x.$$

The proportional arithmetic on ${\mathbb{R}}_{+}$ is obtained by transporting the ordinary arithmetic of $\mathbb{R}$ through $\varphi$. Thus, for $x,y\in {\mathbb{R}}_{+}$, we define

$$x\oplus y:={\varphi }^{-1}\left(\varphi \left(x\right)+\varphi \left(y\right)\right)=xy,$$

(1)

$$x\ominus y:={\varphi }^{-1}\left(\varphi \left(x\right)-\varphi \left(y\right)\right)={\displaystyle \frac{x}{y}},$$

(2)

$$x\odot y:={\varphi }^{-1}\left(\varphi \left(x\right)\varphi \left(y\right)\right)=\exp \left(\left(\ln x\right)\left(\ln y\right)\right)=x^{\ln y},$$

(3)

and, whenever $y\ne 1$,

$$x\oslash y:={\varphi }^{-1}\left({\displaystyle \frac{\varphi \left(x\right)}{\varphi \left(y\right)}}\right)=\exp \left({\displaystyle \frac{\ln x}{\ln y}}\right)=x^{1/\ln y}.$$

(4)

The operation ⊕ plays the role of addition in the proportional arithmetic, whereas ⊙ plays the role of multiplication. The neutral element for ⊕ is 1, while the neutral element for ⊙ is e. Thus, in proportional arithmetic, 1 is the proportional zero and e is the proportional unit.

Remark 1. 

The terminology “proportional addition” and “proportional multiplication” should not be interpreted in terms of the usual arithmetic symbols. In the proportional setting, the operation ⊕ is the addition induced by the logarithmic representation, even though it is realized as the ordinary product $xy$. Similarly, ⊙ is the multiplication induced by the same representation, even though it is realized as $x^{\ln y}$.

Proposition 2 

(Real proportional field). The structure $({\mathbb{R}}_{+},\oplus ,\odot )$ is a field. More precisely, the map

$$\varphi :{\mathbb{R}}_{+}\to \mathbb{R},\varphi \left(x\right)=\ln x,$$

is a field isomorphism from $({\mathbb{R}}_{+},\oplus ,\odot )$ onto $(\mathbb{R},+,·)$.

Proof. 

For $x,y\in {\mathbb{R}}_{+}$, one has

$$\varphi (x\oplus y)=\varphi \left(x\right)+\varphi \left(y\right),\varphi (x\odot y)=\varphi \left(x\right)\varphi \left(y\right).$$

Since $\varphi$ is bijective and $(\mathbb{R},+,·)$ is a field, all field axioms for $({\mathbb{R}}_{+},\oplus ,\odot )$ follow by transport through $\varphi$. The additive identity in the proportional field is ${\varphi }^{-1}\left(0\right)=1$, and the multiplicative identity is ${\varphi }^{-1}\left(1\right)=e$. □

Remark 2 

(Commutativity of the proportional product). Although the ordinary formula $x\odot y=x^{\ln y}$ is not visually symmetric, the operation is commutative because

$$x\odot y=\exp \left(\right(\ln x\left)\right(\ln y\left)\right)=\exp \left(\right(\ln y\left)\right(\ln x\left)\right)=y\odot x.$$

This property is part of the transported field structure and is used throughout the paper.

The proportional opposite of $x\in {\mathbb{R}}_{+}$ with respect to ⊕ is

$$\ominus x:=1\ominus x={\displaystyle \frac{1}{x}},$$

because $x\oplus (\ominus x)=1$. Similarly, if $x\ne 1$, the proportional inverse of x with respect to ⊙ is

$$x^{\{-1\}}:=e^{1/\ln x},$$

because $x\odot x^{\{-1\}}=e$.

For $n\in \mathbb{N}$, the proportional power of x is defined recursively by

$$x^{\left\{n\right\}}=\underset{n\mathrm{times}}{\underbrace{x\odot x\odot \dots \odot x}}.$$

Using the logarithmic representation, one obtains

$$x^{\left\{n\right\}}=\exp \left({\left(\ln x\right)}^n\right).$$

(5)

Indeed, $\varphi \left(x^{\left\{n\right\}}\right)={\left(\varphi \left(x\right)\right)}^n={\left(\ln x\right)}^n$. We also set

$$x^{\left\{0\right\}}:=e,$$

so that $\varphi \left(x^{\left\{0\right\}}\right)=1$, consistently with the transported multiplicative identity.

Remark 3 

(Notation for proportional powers). The braces in $x^{\left\{n\right\}}$ distinguish powers with respect to the proportional product ⊙ from ordinary powers $x^n$. In particular, $x^{\{-1\}}$ denotes the inverse for ⊙, not the ordinary reciprocal $x^{-1}$.

The logarithmic isomorphism provides a natural way to translate classical algebraic identities into proportional identities. For example, the classical identity ${(a+b)}^2=a^2+2ab+b^2$ has the proportional counterpart

$${(x\oplus y)}^{\left\{2\right\}}=x^{\left\{2\right\}}\oplus (e^2\odot x\odot y)\oplus y^{\left\{2\right\}}.$$

Indeed, after applying $\varphi$, this becomes

$${(\ln x+\ln y)}^2={\left(\ln x\right)}^2+2\left(\ln x\right)\left(\ln y\right)+{\left(\ln y\right)}^2.$$

More generally, the classical binomial formula induces

$${(x\oplus y)}^{\left\{n\right\}}={\displaystyle \underset{k=0}{\overset{n}{⨁}}}\left(e^{\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}\odot x^{\{n-k\}}\odot y^{\left\{k\right\}}\right),$$

(6)

where ⨁ denotes repeated proportional addition; that is, ${⨁}_{k=0}^n{A}_k$ means the proportional sum $A_0\oplus A_1\oplus \cdots \oplus A_n$. All identities of this type are meant in the transported-field sense: applying $\varphi$ converts them exactly into the corresponding classical identities.

Definition 1 

(Proportional absolute value). For $x\in {\mathbb{R}}_{+}$, the proportional absolute value of x is defined by

$${\left|x\right|}_p:=\exp \left(\left|\ln x\right|\right).$$

Equivalently,

$${\left|x\right|}_p=\left\{\begin{array}{cc}x,\hfill & x\ge 1,\hfill \\ {\displaystyle \frac{1}{x}},\hfill & 0<x<1.\hfill \end{array}\right.$$

This definition is the natural analogue of the usual absolute value under the logarithmic representation, since ${\ln \left|x\right|}_p=\left|\ln x\right|$. Thus, the proportional absolute value measures the distance of x from the proportional zero 1 in logarithmic coordinates.

Proposition 3 

(Basic properties of the proportional absolute value). For all $x,y\in {\mathbb{R}}_{+}$,

$${\left|x\right|}_p\ge 1,{\left|x\right|}_p=1⟺x=1,$$

and

$${|x\odot y|}_p={\left|x\right|}_p\odot {\left|y\right|}_p=\exp \left(\left|\right(\ln x\left)\right(\ln y\left)\right|\right).$$

Moreover, ${\ln \left|x\right|}_p=\left|\ln x\right|$.

Proof. 

The first two assertions follow from ${\left|x\right|}_p=\exp \left(\right|\ln x\left|\right)$. Also,

$${|x\odot y|}_p=\exp \left(\left|\ln \right(x\odot y\left)\right|\right)=\exp \left(\left|\right(\ln x\left)\right(\ln y\left)\right|\right).$$

□

The previous discussion shows that proportional arithmetic is not an arbitrary symbolic modification of classical arithmetic. It is a field structure induced on ${\mathbb{R}}_{+}$ by a bijection and therefore belongs naturally to the general class of non-Diophantine arithmetics. This algebraic structure will serve as the foundation for the proportional derivative, the proportional integral, the proportional Fourier transform and the proportional function spaces introduced in the following sections.

3. Proportional Functions, Derivatives and Integrals

In this section we introduce the basic analytic objects associated with proportional arithmetic. Once the proportional operations $\oplus ,\ominus ,\odot ,\oslash$ have been fixed, it is natural to define proportional functions, proportional derivatives and proportional integrals in a way that is compatible with the algebraic structure developed in Section 2. These objects provide the analytic foundation for the proportional Fourier transform.

3.1. Proportional Functions and Representatives

Let $f_c:\mathbb{R}\to \mathbb{R}$ be a classical function. Its associated proportional function is defined on ${\mathbb{R}}_{+}$ by

$$f_p\left(t\right)=\exp \left(f_c\left(\ln t\right)\right),t>0.$$

(7)

In this case, $f_p\left(t\right)>0$ for all $t>0$. Conversely, if $f_p:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is a positive proportional function, its associated classical representative is

$$f_c\left(u\right)=\ln \left(f_p\left(e^u\right)\right),u\in \mathbb{R}.$$

(8)

Remark 4. 

The notation $f_c$ will be used for classical representatives, while $f_p$ will denote the corresponding proportional functions. This distinction is important because proportional differentiation and proportional integration act naturally on $f_p$, whereas their evaluation can often be reduced to the corresponding classical operation on $f_c$.

Example 1 

(Basic proportional functions). The proportional sine and cosine functions are defined by

$${\sin }_p\left(t\right):=\exp \left(\sin \left(\ln t\right)\right),{\cos }_p\left(t\right):=\exp \left(\cos \left(\ln t\right)\right),t>0.$$

They are the proportional functions associated with the classical functions $\sin u$ and $\cos u$, respectively. Similarly,

$$t^{\left\{n\right\}}=\exp \left({\left(\ln t\right)}^n\right)$$

is the proportional function associated with the classical monomial $u^n$.

3.2. Proportional Derivative

Since the proportional zero is 1, increments must approach 1, not 0. Moreover, the difference quotient must be written using proportional subtraction and proportional division.

Definition 2 

(Proportional derivative). Let $I\subset {\mathbb{R}}_{+}$ be an interval and let $f_p:I\to {\mathbb{R}}_{+}$. The proportional derivative of $f_p$ at $x\in I$ is defined by

$$D_p{f}_p\left(x\right):={\displaystyle \frac{d_{\odot }{f}_p}{d_{\odot }x}}\left(x\right):=\underset{\begin{array}{c}h\to 1\\ h\ne 1\end{array}}{lim}\left(f_p(x\oplus h)\ominus f_p\left(x\right)\right)\oslash h,$$

(9)

provided that the limit exists in ${\mathbb{R}}_{+}$.

Since $x\oplus h=xh$, this is equivalently

$$D_p{f}_p\left(x\right)=\underset{\begin{array}{c}h\to 1\\ h\ne 1\end{array}}{lim}\left({\displaystyle \frac{f_p\left(xh\right)}{f_p\left(x\right)}}\right)\oslash h.$$

(10)

Theorem 1 

(Evaluation theorem for proportional derivatives). Let $f_c:\mathbb{R}\to \mathbb{R}$ be differentiable and let

$$f_p\left(x\right)=\exp \left(f_c\left(\ln x\right)\right),x>0.$$

Then $f_p$ is proportionally differentiable and

$$D_p{f}_p\left(x\right)=\exp \left(f_c^{\prime }\left(\ln x\right)\right),x>0.$$

(11)

Proof. 

Using (10),

$${\displaystyle \frac{f_p\left(xh\right)}{f_p\left(x\right)}}=\exp \left(f_c(\ln x+\ln h)-f_c\left(\ln x\right)\right).$$

Therefore,

$$\left({\displaystyle \frac{f_p\left(xh\right)}{f_p\left(x\right)}}\right)\oslash h=\exp \left({\displaystyle \frac{f_c(\ln x+\ln h)-f_c\left(\ln x\right)}{\ln h}}\right).$$

Letting $u=\ln h$, we have $u\to 0$ as $h\to 1$. Hence, the limit is $\exp \left(f_c^{\prime }\left(\ln x\right)\right)$. □

Remark 5 

(Representative form of the proportional derivative). If $f_p$ is differentiable in the ordinary sense and positive, then the representative of its proportional derivative is

$$\ln \left(D_p{f}_p\left(x\right)\right)={\displaystyle \frac{d}{d\ln x}}\ln f_p\left(x\right)={\displaystyle \frac{x{f}_p^{\prime }\left(x\right)}{f_p\left(x\right)}}.$$

Thus, $D_p{f}_p$ is the proportional number whose representative is the logarithmic-rate derivative. More generally, for a sufficiently smooth representative, higher proportional derivatives are defined by

$$\ln \left(D_p^k{f}_p\left(x\right)\right)={\displaystyle \frac{d^k}{d{\left(\ln x\right)}^k}}\ln f_p\left(x\right),k\in \mathbb{N}.$$

This convention keeps the derivative inside proportional arithmetic while making clear which classical derivative is being transported. For nonsmooth representatives, and in particular in Sobolev spaces, proportional derivatives will be understood in the transported weak sense whenever the corresponding classical derivative of ${\mathcal{R}}_p{f}_p$ exists only distributionally.

Corollary 1 

(Basic proportional derivatives). The following identities hold:

$$D_{p}c=1,c\in {\mathbb{R}}_{+},$$

$$D_{p}x=e,$$

and, for $n\in \mathbb{N}$,

$$D_p{x}^{\left\{n\right\}}=e^n\odot x^{\{n-1\}}.$$

For the limiting algebraic case $n=0$, one has $x^{\left\{0\right\}}=e$, the proportional unit; hence, $D_p{x}^{\left\{0\right\}}=D_{p}e=1$, consistently with the derivative of a proportional constant.

Proof. 

These identities follow from Theorem 1 applied to the classical representatives 0, u and $u^n$, respectively. The case $n=0$ is the constant representative 1, whose derivative is 0; after transport this gives the proportional zero 1. □

The usual rules of differentiation have proportional analogues. They follow from the evaluation theorem and the corresponding classical rules.

Proposition 4 

(Linearity rule). Let $f_p,g_p:I\to {\mathbb{R}}_{+}$ be proportionally differentiable and let $a,b\in {\mathbb{R}}_{+}$. Then

$$D_p(a\odot f_p\oplus b\odot g_p)=a\odot D_p{f}_p\oplus b\odot D_p{g}_p.$$

Proof. 

The associated representative of $a\odot f_p\oplus b\odot g_p$ is $\left(\ln a\right)f_c+\left(\ln b\right)g_c$. Differentiating and transporting the identity back to proportional notation gives the result. □

Proposition 5 

(Product rule). Let $f_p,g_p:I\to {\mathbb{R}}_{+}$ be proportionally differentiable. Then

$$D_p(f_p\odot g_p)=(D_p{f}_p\odot g_p)\oplus (f_p\odot D_p{g}_p).$$

Proof. 

The representative of $f_p\odot g_p$ is $f_c{g}_c$. The conclusion follows from ${\left(f_c{g}_c\right)}^{\prime }=f_c^{\prime }{g}_c+f_c{g}_c^{\prime }$. □

Proposition 6 

(Quotient rule). Let $f_p,g_p:I\to {\mathbb{R}}_{+}$ be proportionally differentiable and suppose that $g_p\left(x\right)\ne 1$. Then

$$D_p(f_p\oslash g_p)=\left(D_p{f}_p\odot g_p\ominus f_p\odot D_p{g}_p\right)\oslash g_p^{\left\{2\right\}}.$$

Proof. 

This is the transport of the classical quotient rule for $f_c/g_c$. □

Proposition 7 

(Chain rule). Let $g_p:I\to J$ and $f_p:J\to {\mathbb{R}}_{+}$ be proportionally differentiable. Then

$$D_p(f_p\circ g_p)\left(x\right)=D_p{f}_p\left(g_p\left(x\right)\right)\odot D_p{g}_p\left(x\right),$$

whenever the composition is well defined.

Proof. 

Let $g_c\left(u\right)=\ln \left(g_p\left(e^u\right)\right)$ and $f_c\left(v\right)=\ln \left(f_p\left(e^v\right)\right)$. The representative of $f_p\circ g_p$ is $f_c\left(g_c\left(u\right)\right)$. Differentiating gives $(f_c^{\prime }\circ g_c)g_c^{\prime }$, and transporting this identity back through the proportional arithmetic gives the stated proportional chain rule. □

3.3. Proportional Integral

Definition 3 

(Proportional integral). Let $f_p:(0,\infty )\to {\mathbb{R}}_{+}$ be associated with $f_c:\mathbb{R}\to \mathbb{R}$, that is,

$$f_p\left(t\right)=\exp \left(f_c\left(\ln t\right)\right).$$

If $f_c\in L^1\left(\mathbb{R}\right)$, define

$${\int }_0^{\infty }{f}_p\left(t\right)d_{\odot }t:=\exp \left({\int }_{-\infty }^{\infty }{f}_c\left(u\right)du\right).$$

(12)

More generally, if $0<a<b<\infty$, define

$${\int }_a^b{f}_p\left(t\right)d_{\odot }t:=\exp \left({\int }_{\ln a}^{\ln b}{f}_c\left(u\right)du\right).$$

(13)

Remark 6 

(Finite and extended proportional integrals). Definition 3 is a finite ${\mathbb{R}}_{+}$-valued integral and therefore assumes that the representative integral is a finite real number. If an improper representative integral equals $-\infty$, its exponential limit is 0, which does not belong to ${\mathbb{R}}_{+}=(0,\infty )$. Such a value may be recorded only in the extended closure $[0,\infty ]$; it is not an element of the proportional field used in the algebraic results below. The Fourier theory in this paper uses finite complex integrals of representatives and is therefore unaffected by this extended-value convention.

Equivalently,

$${\int }_a^b{f}_p\left(t\right)d_{\odot }t=\exp \left({\int }_a^b\ln \left(f_p\left(t\right)\right){\displaystyle \frac{dt}{t}}\right).$$

(14)

Proposition 8 

(Linearity of the proportional integral). Let $f_p,g_p$ be associated with $f_c, g_c\in L^1\left(\mathbb{R}\right)$, respectively, and let $a,b\in {\mathbb{R}}_{+}$. Then

$${\int }_0^{\infty }\left(a\odot f_p\left(t\right)\oplus b\odot g_p\left(t\right)\right)d_{\odot }t=a\odot {\int }_0^{\infty }{f}_p\left(t\right)d_{\odot }t\oplus b\odot {\int }_0^{\infty }{g}_p\left(t\right)d_{\odot }t.$$

Proof. 

The representative of $a\odot f_p\oplus b\odot g_p$ is $\left(\ln a\right)f_c+\left(\ln b\right)g_c$. The result follows from the classical linearity of the integral. □

Theorem 2 

(Fundamental theorem of proportional calculus). Let $F_c:\mathbb{R}\to \mathbb{R}$ be continuously differentiable and define

$$F_p\left(t\right)=\exp \left(F_c\left(\ln t\right)\right),t>0.$$

Then, for $0<a<b<\infty$,

$${\int }_a^b{D}_p{F}_p\left(t\right)d_{\odot }t=F_p\left(b\right)\ominus F_p\left(a\right).$$

Proof. 

By Theorem 1, $D_p{F}_p\left(t\right)=\exp \left(F_c^{\prime }\left(\ln t\right)\right)$. Therefore,

$${\int }_a^b{D}_p{F}_p\left(t\right)d_{\odot }t=\exp \left({\int }_{\ln a}^{\ln b}{F}_c^{\prime }\left(u\right)du\right)=\exp (F_c\left(\ln b\right)-F_c\left(\ln a\right)).$$

This equals $F_p\left(b\right)/F_p\left(a\right)=F_p\left(b\right)\ominus F_p\left(a\right)$. □

3.4. Real Proportional $L^q$ Spaces

Definition 4 

(Real proportional $L^q$ spaces). Let $1\le q\le \infty$. The real proportional space $L_p^q(0,\infty )$ is the set of ${\mathbb{R}}_{+}$-valued proportional functions $f_p$ whose representative $f_c\left(u\right)=\ln \left(f_p\left(e^u\right)\right)$ belongs to $L^q\left(\mathbb{R}\right)$. The norm is defined by

$$\parallel f_p{\parallel }_{L_p^q}:={\parallel f_c\parallel }_{L^q\left(\mathbb{R}\right)}.$$

For $1\le q<\infty$, the proportional magnitude associated with this norm is

$$\exp \left(\parallel f_p{\parallel }_{L_p^q}^q\right)={\int }_0^{\infty }{\left|f_p\left(t\right)\right|}_p^{\left\{q\right\}}{d}_{\odot }t,$$

whenever the right-hand side is interpreted through the representative $|f_c{|}^q$.

Remark 7. 

The norm $\parallel f_p{\parallel }_{L_p^q}$ is an ordinary real norm induced by the representative, not a proportional number. This convention avoids confusing proportional magnitudes with Banach-space norms. The complex-valued spaces required by the proportional Fourier transform are introduced after the proportional complex field has been defined. We retain this real-first order only to motivate the elementary calculus before introducing the formal complex field; Definition 12 is the definitive functional-analytic setting, and the real space is subsequently identified as its real-representative subspace.

4. Proportional Complex Numbers and the Proportional Exponential Kernel

The proportional Fourier transform requires a proportional analogue of the complex exponential kernel. In the classical Fourier transform, the oscillatory factor $e^{-ix\xi }$ is built from the imaginary unit i, the ordinary product $x\xi$, and the classical exponential function. Therefore, in the proportional setting, it is necessary to introduce a coherent notion of proportional complex numbers, a proportional imaginary unit, and a proportional exponential kernel compatible with $\oplus ,\ominus ,\odot ,\oslash$.

To avoid branch ambiguities associated with the multivalued complex logarithm, and in view of the technical issues that arise in complex multiplicative calculus [4,5], we introduce proportional complex numbers as a formal copy of $\mathbb{C}$.

The field ${\mathbb{C}}_p$ is not analytically necessary: every calculation can be carried out entirely with representatives in $\mathbb{C}$. Its role is organizational and semantic. It permits the transform, kernel, scalar multiplication, conjugation, and codomains to be written in the same proportional arithmetic as the original positive-scale model, while ${\mathsf{\ell }}_p$ provides an exact audit map to classical complex analysis. Accordingly, ${\mathbb{C}}_p$ should be read as a formal transported field and not as a source of additional complex-analytic content.

Definition 5 

(Proportional complex field). Let ${\mathbb{C}}_p$ be a set in bijection with $\mathbb{C}$, and fix a bijection

$$\mathcal{E}:\mathbb{C}\to {\mathbb{C}}_p$$

whose restriction to the real axis is identified with the real proportional embedding $\mathcal{E}\left(u\right)=e^u\in {\mathbb{R}}_{+}$, $u\in \mathbb{R}$. Thus, the real proportional field is the embedded subfield $\mathcal{E}\left(\mathbb{R}\right)\subset {\mathbb{C}}_p$, and the notation $e^u$ is used only for this real embedded copy. Let ${\mathsf{\ell }}_p={\mathcal{E}}^{-1}$. For $z,w\in \mathbb{C}$, define

$$\mathcal{E}\left(z\right)\oplus \mathcal{E}\left(w\right)=\mathcal{E}(z+w),\mathcal{E}\left(z\right)\ominus \mathcal{E}\left(w\right)=\mathcal{E}(z-w),$$

$$\mathcal{E}\left(z\right)\odot \mathcal{E}\left(w\right)=\mathcal{E}\left(zw\right),\mathcal{E}\left(z\right)\oslash \mathcal{E}\left(w\right)=\mathcal{E}(z/w),w\ne 0.$$

Proposition 9 

(Complex proportional field). With the operations in Definition 5, $({\mathbb{C}}_p,\oplus ,\odot )$ is a field, and ${\mathsf{\ell }}_p:{\mathbb{C}}_p\to \mathbb{C}$ is a field isomorphism.

Proof. 

Applying ${\mathsf{\ell }}_p$ converts every proportional operation into the corresponding ordinary complex operation. The field axioms follow from the field axioms of $\mathbb{C}$. □

Definition 6 

(Topology, Borel structure, and measurable proportional functions). The topology on ${\mathbb{C}}_p$ is the pullback of the Euclidean topology on $\mathbb{C}$ through ${\mathsf{\ell }}_p$. Thus, $U\subset {\mathbb{C}}_p$ is open if and only if ${\mathsf{\ell }}_p\left(U\right)$ is open in $\mathbb{C}$. Its Borel σ-algebra is the corresponding pullback Borel structure. Consequently,

$$z_{p,n}\to z_p\mathrm{in}{\mathbb{C}}_p⟺{\mathsf{\ell }}_p\left(z_{p,n}\right)\to {\mathsf{\ell }}_p\left(z_p\right)\mathrm{in}\mathbb{C}.$$

Continuity, measurability, almost-everywhere equality, and integrability of ${\mathbb{C}}_p$-valued functions are understood through this identification. In particular, the complex proportional integral is the transported Lebesgue integral of the representative; no independent Bochner theory on an unspecified topology is being assumed.

With the embedded real copy fixed in Definition 5, the previous real proportional arithmetic is a subfield of ${\mathbb{C}}_p$. The proportional zero is $\mathcal{E}\left(0\right)=1$, and the proportional unit is $\mathcal{E}\left(1\right)=e$.

Definition 7 

(Proportional imaginary unit). The proportional imaginary unit is

$$i_p:=\mathcal{E}\left(i\right).$$

It satisfies

$$i_p^{\left\{2\right\}}=i_p\odot i_p=\mathcal{E}\left(i^2\right)=\mathcal{E}(-1)=e^{-1}=\ominus e.$$

Every proportional complex number can be written in the form

$$z_p=a\oplus i_p\odot b,a,b\in {\mathbb{R}}_{+},$$

which corresponds to the classical complex number

$${\mathsf{\ell }}_p\left(z_p\right)=\ln a+i\ln b.$$

Indeed,

$$a\oplus i_p\odot b=\mathcal{E}\left(\ln a\right)\oplus \mathcal{E}\left(i\right)\odot \mathcal{E}\left(\ln b\right)=\mathcal{E}(\ln a+i\ln b).$$

Definition 8 

(Proportional conjugate and modulus). If $z_p=\mathcal{E}\left(z\right)\in {\mathbb{C}}_p$, the proportional conjugate and proportional modulus are defined by

$$z_p^{\star }:=\mathcal{E}\left(\overline{z}\right),|z_p{|}_p:=\mathcal{E}\left(\right|z\left|\right).$$

Thus, the proportional conjugate is transported from the ordinary complex conjugate; it is not obtained by applying the real logarithm to a complex number. Equivalently,

$${\mathsf{\ell }}_p\left(z_p^{\star }\right)=\overline{{\mathsf{\ell }}_p\left(z_p\right)}.$$

In words, the representative of the proportional conjugate is the ordinary complex conjugate of the representative. This textual convention is included to avoid any ambiguity between proportional conjugation in ${\mathbb{C}}_p$ and ordinary real logarithms on ${\mathbb{R}}_{+}$. If $z_p=a\oplus i_p\odot b$, then

$$|z_p{|}_p=\exp \left(\sqrt{{\left(\ln a\right)}^2+{\left(\ln b\right)}^2}\right),$$

and

$$|z_p{|}_p^{\left\{2\right\}}=a^{\left\{2\right\}}\oplus b^{\left\{2\right\}}.$$

Proposition 10 

(Product with the proportional conjugate). For every $z_p\in {\mathbb{C}}_p$,

$$z_p\odot z_p^{\star }={\left|z_p\right|}_p^{\left\{2\right\}}.$$

Proof. 

Let $z_p=\mathcal{E}\left(z\right)$. Then

$$z_p\odot z_p^{\star }=\mathcal{E}\left(z\right)\odot \mathcal{E}\left(\overline{z}\right)=\mathcal{E}\left(z\overline{z}\right)={\mathcal{E}\left(\right|z|}^2{)=\mathcal{E}(\left|z\right|)}^{\left\{2\right\}}={\left|z_p\right|}_p^{\left\{2\right\}}.$$

□

Definition 9 

(Proportional exponential). For $Z_p\in {\mathbb{C}}_p$, the proportional exponential is defined by

$${\mathrm{Exp}}_p\left(Z_p\right):=\mathcal{E}\left(\exp \left({\mathsf{\ell }}_p\left(Z_p\right)\right)\right).$$

To avoid confusing the formal embedding $\mathcal{E}$, the classical exponential exp, and the transported proportional exponential ${\mathrm{Exp}}_p$, we use the notation ${\mathrm{Exp}}_p$ consistently throughout. The proportional Euler identity takes the form

$${\mathrm{Exp}}_p(i_p\odot t)={\cos }_p\left(t\right)\oplus i_p\odot {\sin }_p\left(t\right),t\in {\mathbb{R}}_{+}.$$

Indeed, both sides have classical representative $e^{i\ln t}=\cos \left(\ln t\right)+i\sin \left(\ln t\right)$.

Definition 10 

(Proportional Fourier kernel). For $t,\xi \in {\mathbb{R}}_{+}$, the proportional Fourier kernel is

$$K_p(t,\xi ):={\mathrm{Exp}}_p(\ominus i_p\odot \xi \odot t).$$

Proposition 11 

(Classical representative of the proportional kernel). For $t,\xi \in {\mathbb{R}}_{+}$,

$${\mathsf{\ell }}_p\left(K_p(t,\xi )\right)=e^{-i\left(\ln t\right)\left(\ln \xi \right)}.$$

Proof. 

The proportional product $i_p\odot \xi \odot t$ has representative $i\left(\ln \xi \right)\left(\ln t\right)$. Taking the proportional opposite gives the representative $-i\left(\ln \xi \right)\left(\ln t\right)$. Applying the proportional exponential gives the representative $e^{-i\left(\ln t\right)\left(\ln \xi \right)}$. □

Complex-Valued Proportional Functions and Spaces

After the proportional complex field has been fixed, proportional functions may take values in ${\mathbb{C}}_p$. Their representative is defined by

$${\mathcal{R}}_p{f}_p\left(u\right):={\mathsf{\ell }}_p\left(f_p\left(e^u\right)\right),u\in \mathbb{R}.$$

Conversely, every classical complex-valued function $f_c$ on $\mathbb{R}$ determines the proportional function

$$f_p\left(x\right)=\mathcal{E}\left(f_c\left(\ln x\right)\right),x\in {\mathbb{R}}_{+}.$$

For ${\mathbb{R}}_{+}$-valued functions this agrees with $f_p\left(x\right)=\exp \left(f_c\left(\ln x\right)\right)$. Thus, the representative is a proof device associated with the proportional arithmetic; it does not replace the proportional operations used in the definitions.

Remark 8 

(Notation policy for representatives). We use ${\mathcal{R}}_p{f}_p$ in definitions and abstract statements. At the beginning of a calculation or proof we may set $f_c:={\mathcal{R}}_p{f}_p$ to reduce notation. The pointwise expression ${\mathsf{\ell }}_p\left(f_p\left(t\right)\right)$ is reserved for formulas written directly in the positive variable $t>0$. This convention avoids switching among three equivalent notations without purpose.

Definition 11 

(Complex proportional integral). Let $F_p:(0,\infty )\to {\mathbb{C}}_p$ and let $F_c={\mathcal{R}}_p{F}_p$. If $F_c\in L^1(\mathbb{R};\mathbb{C})$, define

$${\int }_0^{\infty }{F}_p\left(t\right)d_{\odot }t:=\mathcal{E}\left({\int }_{-\infty }^{\infty }{F}_c\left(u\right)du\right).$$

More generally, for $0<a<b<\infty$,

$${\int }_a^b{F}_p\left(t\right)d_{\odot }t:=\mathcal{E}\left({\int }_{\ln a}^{\ln b}{F}_c\left(u\right)du\right).$$

Definition 12 

(Complex proportional $L^q$ spaces). Let $1\le q\le \infty$. We define the space of equivalence classes of measurable ${\mathbb{C}}_p$-valued functions

$$L_p^q({\mathbb{R}}_{+};{\mathbb{C}}_p):=\{\left[f_p\right]:{\mathcal{R}}_p{f}_p\in L^q(\mathbb{R};\mathbb{C})\},$$

with norm

$$\parallel f_p{\parallel }_{L_p^q}:={\parallel {\mathcal{R}}_p{f}_p\parallel }_{L^q\left(\mathbb{R}\right)}.$$

For $q=2$, the induced Hilbert-space inner product is

$${\langle f_p,g_p\rangle }_{p,2}:={\int }_{\mathbb{R}}{\mathcal{R}}_p{f}_p\left(u\right)\overline{{\mathcal{R}}_p{g}_p\left(u\right)}du.$$

Here the bar denotes the ordinary complex conjugate of the representative. Equivalently, conjugation is applied after the proportional function has been transported by ${\mathcal{R}}_p$. This convention is essential: the Hilbert structure is transported from the classical complex Hilbert space and is not obtained by multiplying representatives without conjugation. The real proportional space $L_p^q(0,\infty )$ is identified with the subspace of $L_p^q({\mathbb{R}}_{+};{\mathbb{C}}_p)$ whose representative is real-valued.

The Hilbert inner product is deliberately an ordinary complex number, transported from $L^2(\mathbb{R};\mathbb{C})$. A ${\mathbb{C}}_p$-valued expression could be defined by $\mathcal{E}\left({\langle {\mathcal{R}}_p{f}_p,{\mathcal{R}}_p{g}_p\rangle }_{L^2}\right)$, but it would not be a Hilbert-space inner product in the standard sense because positivity and scalar linearity would then be expressed in the transported field rather than in $\mathbb{C}$. We therefore keep proportional magnitudes for algebraic interpretation and use the classical scalar-valued inner product for functional analysis.

Throughout the paper, equalities involving $L_p^2$ functions or proportional Sobolev functions are understood at the level of equivalence classes; equivalently, after representatives are chosen, they hold almost everywhere in the logarithmic variable.

Proposition 12 

(Representative isometry and transport principle). For every $1\le q\le \infty$, the representative map

$${\mathcal{R}}_p:L_p^q({\mathbb{R}}_{+};{\mathbb{C}}_p)⟶L^q(\mathbb{R};\mathbb{C}),{\mathcal{R}}_p{f}_p\left(u\right)={\mathsf{\ell }}_p\left(f_p\left(e^u\right)\right),$$

is an isometric isomorphism with inverse ${\mathcal{R}}_p^{-1}{f}_c\left(x\right)=\mathcal{E}\left(f_c\left(\ln x\right)\right)$. In particular, $L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$ is a Hilbert space with the inner product of Definition 12. More generally, if X and Y are classical function spaces and $X_p={\mathcal{R}}_p^{-1}\left(X\right)$, $Y_p={\mathcal{R}}_p^{-1}\left(Y\right)$, then any operator $T:X\to Y$ induces a proportional operator

$$T_p:={\mathcal{R}}_p^{-1}T{\mathcal{R}}_p:X_p\to Y_p.$$

Algebraic identities, norm estimates and continuity properties of T transfer to $T_p$ whenever the corresponding classical hypotheses hold. This principle is used only as a verification mechanism; the definitions of the proportional operators below remain written in the proportional algebra.

Proof. 

The first assertion follows directly from the definition of $L_p^q({\mathbb{R}}_{+};{\mathbb{C}}_p)$ and of its norm. The formula for the inverse is immediate from ${\mathsf{\ell }}_p\left(\mathcal{E}\left(z\right)\right)=z$. For the transport statement, apply ${\mathcal{R}}_p$ to the proposed proportional identity or estimate; it becomes the corresponding classical identity or estimate for T. Applying ${\mathcal{R}}_p^{-1}$ returns the result to the proportional setting. □

In subsequent sections, whenever a proportional statement is exactly the transport of a classical theorem through ${\mathcal{R}}_p$, the proof identifies the representative statement and then applies Proposition 12.

Definition 13 

(Complex proportional derivative). Let $I\subset {\mathbb{R}}_{+}$ and let $f_p:I\to {\mathbb{C}}_p$. If the representative $f_c={\mathcal{R}}_p{f}_p$ is differentiable on $\ln I$, the proportional derivative of $f_p$ is

$$D_p{f}_p\left(x\right):=\mathcal{E}\left(f_c^{\prime }\left(\ln x\right)\right),x\in I.$$

Higher derivatives are defined by $D_p^k{f}_p\left(x\right)=\mathcal{E}\left(f_c^{\left(k\right)}\left(\ln x\right)\right)$ whenever the representative derivatives exist. For ${\mathbb{R}}_{+}$-valued functions this definition agrees with the limit definition in Section 3.

Remark 9 

(Complex proportional differentiation rules). The proportional differentiation rules stated for ${\mathbb{R}}_{+}$-valued functions extend to ${\mathbb{C}}_p$-valued functions through Definition 13, because their representatives satisfy the corresponding classical complex differentiation rules. This extension keeps the algebraic notation proportional while using representatives only to verify the rule.

The following dictionary summarizes the representative identities used later. It is not a replacement for the proportional formulation; it is the verification mechanism ensuring that proportional statements have precise classical counterparts:

$$\begin{array}{cc}\mathrm{Proportional}\mathrm{object}& \mathrm{Representative}\mathrm{identity}\\ f_p\left(x\right)& {\mathcal{R}}_p{f}_p\left(u\right)={\mathsf{\ell }}_p\left(f_p\left(e^u\right)\right)\\ D_p{f}_p\left(x\right)& {\left({\mathcal{R}}_p{f}_p\right)}^{\prime }\left(u\right)\\ {\displaystyle {\int }_0^{\infty }{f}_p\left(t\right)d_{\odot }t}& {\displaystyle {\int }_{-\infty }^{\infty }{\mathcal{R}}_p{f}_p\left(u\right)du}\\ K_p(t,\xi )& e^{-i\left(\ln t\right)\left(\ln \xi \right)}\\ {\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)& \mathcal{F}\left\{{\mathcal{R}}_p{f}_p\right\}\left(\ln \xi \right)\end{array}$$

The proportional complex structure and the proportional exponential kernel introduced in this section will be used in the next section to define the proportional Fourier transform. The key idea is that the classical oscillatory factor $e^{-ix\xi }$ is replaced by $K_p(t,\xi )={\mathrm{Exp}}_p(\ominus i_p\odot \xi \odot t)$, which is expressed entirely in terms of proportional objects.

5. The Proportional Fourier Transform

We now introduce the proportional Fourier transform. The construction follows the structure of the classical Fourier transform, but each ingredient is replaced by its proportional counterpart: the proportional normalization factor, the proportional integral, proportional multiplication and the proportional exponential kernel. The representative results in this section are used to prove rigorously that the proportional operator has the expected analytic properties.

5.1. Definition and Well-Definedness

Throughout this paper we use the symmetric Fourier normalization. In the proportional setting, the classical coefficient $1/\sqrt{2\pi }$ is represented by

$${\eta }_p:=\mathcal{E}\left({\displaystyle \frac{1}{\sqrt{2\pi }}}\right)=\exp \left({\displaystyle \frac{1}{\sqrt{2\pi }}}\right).$$

Definition 14 

(Proportional Fourier transform). Let $f_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$. The proportional Fourier transform of $f_p$ is

$${\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right):={\eta }_p\odot {\int }_0^{\infty }{f}_p\left(t\right)\odot K_p(t,\xi )d_{\odot }t,\xi \in {\mathbb{R}}_{+},$$

(15)

where

$$K_p(t,\xi )={\mathrm{Exp}}_p(\ominus i_p\odot \xi \odot t).$$

We also write

$${\widehat{f_p}}^p\left(\xi \right)={\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right).$$

The definition is stated entirely in proportional notation. The complex proportional integral in Definition 11 ensures that the integral in (15) is meaningful whenever the representative is integrable.

Proposition 13 

(Well-definedness). Let $f_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$, and let $f_c={\mathcal{R}}_p{f}_p\in L^1(\mathbb{R};\mathbb{C})$. Then ${\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)\in {\mathbb{C}}_p$ is well defined for every $\xi >0$.

Proof. 

For fixed $\xi >0$, the integrand $f_p\left(t\right)\odot K_p(t,\xi )$ has representative

$$f_c\left(u\right)e^{-iu\ln \xi },u=\ln t.$$

Since $|e^{-iu\ln \xi }|=1$ and $f_c\in L^1\left(\mathbb{R}\right)$, this representative belongs to $L^1(\mathbb{R};\mathbb{C})$. Hence, the proportional integral exists and its value belongs to ${\mathbb{C}}_p$. □

5.2. Correspondence, Continuity and Inversion

Theorem 3 

(Correspondence theorem). Let $f_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$, and let $f_c={\mathcal{R}}_p{f}_p\in L^1(\mathbb{R};\mathbb{C})$. Then, for every $\xi >0$,

$${\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)=\mathcal{E}\left(\mathcal{F}\left\{f_c\right\}\left(\ln \xi \right)\right),$$

(16)

where

$$\mathcal{F}\left\{f_c\right\}\left(\omega \right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{\infty }{f}_c\left(u\right)e^{-iu\omega }du.$$

Equivalently,

$${\mathcal{R}}_p\left({\mathcal{F}}_p\left\{f_p\right\}\right)\left(\omega \right)=\mathcal{F}\left\{{\mathcal{R}}_p{f}_p\right\}\left(\omega \right),\omega \in \mathbb{R}.$$

(17)

If $\mathcal{F}\left\{f_c\right\}\left(\ln \xi \right)\in \mathbb{R}$, the proportional value may be identified with the positive real number $\exp \left(\mathcal{F}\left\{f_c\right\}\left(\ln \xi \right)\right)$. No such real-valued conclusion is implied for a general real-valued $f_c$.

Proof. 

The representative of the proportional integral in (15) is

$${\int }_{-\infty }^{\infty }{f}_c\left(u\right)e^{-iu\ln \xi }du.$$

Multiplication by ${\eta }_p=\mathcal{E}(1/\sqrt{2\pi })$ in the proportional sense multiplies this representative by $1/\sqrt{2\pi }$. Hence, the representative of ${\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)$ is $\mathcal{F}\left\{f_c\right\}\left(\ln \xi \right)$, which proves (16) and (17). □

Remark 10 

(Haar measure and Mellin-type expression). The correspondence theorem also gives a useful way to compare ${\mathcal{F}}_p$ with Mellin-type analysis without changing the definition of the proportional transform. If $f_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$, then

$${\mathsf{\ell }}_p\left({\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_0^{\infty }{\mathsf{\ell }}_p\left(f_p\left(t\right)\right)t^{-i\ln \xi }{\displaystyle \frac{dt}{t}}.$$

Thus, the Haar measure $dt/t$ on ${\mathbb{R}}_{+}$ appears after taking representatives. The proportional transform is nevertheless not the ordinary Mellin transform of $f_p$: the integrand is the representative ${\mathsf{\ell }}_p\left(f_p\left(t\right)\right)$, and the value is transported back to ${\mathbb{C}}_p$.

Proposition 14 

(Exact relation with the group Fourier and Mellin transforms). Let $G=(0,\infty )$ with multiplication and Haar measure $dt/t$, and define

$$g\left(t\right):={\mathsf{\ell }}_p\left(f_p\left(t\right)\right).$$

For $\omega \in \mathbb{R}$, the functions ${\chi }_{\omega }\left(t\right)=t^{-i\omega }$ are the unitary characters of G. With the symmetric normalization, the group Fourier transform of g is

$${\mathcal{F}}_{G}g\left(\omega \right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_0^{\infty }g\left(t\right)t^{-i\omega }{\displaystyle \frac{dt}{t}}.$$

Then

$${\mathsf{\ell }}_p\left({\mathcal{F}}_p\left\{f_p\right\}\left(e^{\omega }\right)\right)={\mathcal{F}}_{G}g\left(\omega \right).$$

Moreover, if the Mellin transform is defined by

$$\mathcal{M}g\left(s\right)={\int }_0^{\infty }g\left(t\right)t^{s-1}dt,$$

then, whenever both sides exist,

$${\mathsf{\ell }}_p\left({\mathcal{F}}_p\left\{f_p\right\}\left(e^{\omega }\right)\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}\mathcal{M}g(-i\omega ).$$

Thus, the representative of the proportional transform is exactly the Fourier transform on the multiplicative locally compact abelian group, equivalently a Mellin transform on the imaginary axis. The transformed object is $g={\mathsf{\ell }}_p\circ f_p$, not the ordinary values of $f_p$.

Proof. 

The first identity follows by the change of variables $t=e^u$, for which $dt/t=du$, and by the character identity $t^{-i\omega }=e^{-iu\omega }$. The Mellin identity follows from $t^{s-1}dt=t^{s}dt/t$ with $s=-i\omega$. □

Definition 15 

(Proportional vanishing at infinity). A function $G_p:{\mathbb{R}}_{+}\to {\mathbb{C}}_p$ belongs to $C_{0,p}({\mathbb{R}}_{+};{\mathbb{C}}_p)$ if ${\mathcal{R}}_p{G}_p\in C_0(\mathbb{R};\mathbb{C})$. Equivalently, $G_p$ is continuous in the proportional topology and

$$G_p\left(\xi \right)⟶\mathcal{E}\left(0\right)=1\mathrm{as}\left|\ln \xi \right|\to \infty .$$

Thus, “vanishing” means convergence to the proportional zero 1, not to the ordinary number 0.

Theorem 4 

(Riemann–Lebesgue property). The proportional Fourier transform maps

$${\mathcal{F}}_p:L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)⟶C_{0,p}({\mathbb{R}}_{+};{\mathbb{C}}_p).$$

Equivalently, if $f_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$, then ${\mathcal{F}}_p\left\{f_p\right\}\in C_{0,p}({\mathbb{R}}_{+};{\mathbb{C}}_p)$.

Proof. 

By Theorem 3, ${\mathcal{R}}_p\left({\mathcal{F}}_p\left\{f_p\right\}\right)=\mathcal{F}\left\{{\mathcal{R}}_p{f}_p\right\}$. Since ${\mathcal{R}}_p{f}_p\in L^1(\mathbb{R};\mathbb{C})$, the classical Riemann–Lebesgue lemma gives $\mathcal{F}\left\{{\mathcal{R}}_p{f}_p\right\}\in C_0(\mathbb{R};\mathbb{C})$. Transporting this statement gives the result. □

Definition 16 

(Inverse proportional Fourier transform). Let $G_p:{\mathbb{R}}_{+}\to {\mathbb{C}}_p$ be a proportional function for which the following proportional integral exists. The inverse proportional Fourier transform is

$${\mathcal{F}}_p^{-1}\left\{G_p\right\}\left(t\right):={\eta }_p\odot {\int }_0^{\infty }{G}_p\left(\xi \right)\odot {\mathrm{Exp}}_p(i_p\odot \xi \odot t)d_{\odot }\xi ,t>0.$$

(18)

Theorem 5 

(Inversion theorem). Let $f_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$ and set $f_c={\mathcal{R}}_p{f}_p$. Suppose only that $\mathcal{F}\left\{f_c\right\}\in L^1(\mathbb{R};\mathbb{C})$; the condition $f_c\in L^1$ is already contained in $f_p\in L_p^1$. Then, at every point $t>0$ at which the classical Fourier inversion formula holds for $f_c$,

$$f_p\left(t\right)={\mathcal{F}}_p^{-1}\left\{{\mathcal{F}}_p\left\{f_p\right\}\right\}\left(t\right).$$

Proof. 

The representative of ${\mathcal{F}}_p\left\{f_p\right\}$ is $\mathcal{F}\left\{f_c\right\}$. Therefore, the representative of the inverse proportional transform is

$${\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{\infty }\mathcal{F}\left\{f_c\right\}\left(\omega \right)e^{i\omega \ln t}d\omega ,$$

which equals $f_c\left(\ln t\right)$ by the classical Fourier inversion theorem. Transporting this equality back gives $f_p\left(t\right)$. □

5.3. Plancherel Theorem and $L^2$-Extension

Theorem 6 

(Proportional Plancherel theorem). For every $f_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)\cap L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$,

$$\parallel {\mathcal{F}}_p\left\{f_p\right\}{\parallel }_{L_p^2}={\parallel f_p\parallel }_{L_p^2}.$$

More generally, for $f_p,g_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)\cap L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$,

$${\langle {\mathcal{F}}_p\left\{f_p\right\},{\mathcal{F}}_p\left\{g_p\right\}\rangle }_{p,2}={\langle f_p,g_p\rangle }_{p,2}.$$

Consequently, ${\mathcal{F}}_p$ extends uniquely to a unitary operator on $L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$.

Proof. 

Let $f_c={\mathcal{R}}_p{f}_p$ and $g_c={\mathcal{R}}_p{g}_p$. By Theorem 3, the representative of ${\mathcal{F}}_p\left\{f_p\right\}$ is $\mathcal{F}\left\{f_c\right\}$. The classical Plancherel theorem for the symmetric Fourier transform [14,15] gives

$$\parallel \mathcal{F}\left\{f_c\right\}{\parallel }_{L^2\left(\mathbb{R}\right)}={\parallel f_c\parallel }_{L^2\left(\mathbb{R}\right)},{\langle \mathcal{F}\left\{f_c\right\},\mathcal{F}\left\{g_c\right\}\rangle }_{L^2}={\langle f_c,g_c\rangle }_{L^2}.$$

Using the transported norm and inner product in Definition 12, together with Proposition 12, gives the stated identities. Since $L^1(\mathbb{R};\mathbb{C})\cap L^2(\mathbb{R};\mathbb{C})$ is dense in $L^2(\mathbb{R};\mathbb{C})$, the representative isomorphism implies that $L_p^1\cap L_p^2$ is dense in $L_p^2$. If U denotes the classical unitary Fourier operator on $L^2(\mathbb{R};\mathbb{C})$, the transported operator ${\mathcal{R}}_p^{-1}U{\mathcal{R}}_p$ is unitary on $L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$ and agrees with the integral definition of ${\mathcal{F}}_p$ on the dense subspace $L_p^1\cap L_p^2$. Hence, the extension is unique and coincides with this transported unitary operator. □

5.4. Interpretation

The precise scope of the construction is already stated in Proposition 1: ${\mathcal{F}}_p$ is internally formulated with proportional operations, while its analytic content is controlled by conjugation with ${\mathcal{R}}_p$. We therefore do not repeat that positioning in each transported result.

6. Operational Properties of the Proportional Fourier Transform

In this section we establish the main operational properties of the proportional Fourier transform. These properties show that the transform introduced in Section 5 preserves, within proportional arithmetic, the structural behavior of the classical Fourier transform. In particular, we prove proportional analogues of linearity, translation, modulation, scaling, differentiation and convolution.

Throughout this section, $f_p,g_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$ denote proportional functions with representatives $f_c,g_c\in L^1(\mathbb{R};\mathbb{C})$, respectively. The rules are written in proportional notation; each proof verifies the corresponding representative identity.

6.1. Linearity

Proposition 15 

(Linearity). Let $f_p,g_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$ and let $a,b\in {\mathbb{C}}_p$. Then

$${\mathcal{F}}_p\{a\odot f_p\oplus b\odot g_p\}\left(\xi \right)=a\odot {\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)\oplus b\odot {\mathcal{F}}_p\left\{g_p\right\}\left(\xi \right),\xi >0.$$

Proof. 

The representative of $a\odot f_p\oplus b\odot g_p$ is ${\mathsf{\ell }}_p\left(a\right)f_c+{\mathsf{\ell }}_p\left(b\right)g_c$. The result follows from the classical linearity of $\mathcal{F}$ and Theorem 3. □

6.2. Translation and Modulation

Proposition 16 

(Proportional translation). Let $t_0\in {\mathbb{R}}_{+}$ and define $h_p\left(t\right):=f_p(t\ominus t_0)$. Then

$${\mathcal{F}}_p\left\{h_p\right\}\left(\xi \right)=K_p(t_0,\xi )\odot {\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right),\xi >0.$$

Proof. 

The representative of $h_p$ is $h_c\left(u\right)=f_c(u-\ln t_0)$. The classical translation rule gives

$$\mathcal{F}\left\{h_c\right\}\left(\omega \right)=e^{-i\omega \ln t_0}\mathcal{F}\left\{f_c\right\}\left(\omega \right).$$

At $\omega =\ln \xi$, the factor $e^{-i\left(\ln \xi \right)\left(\ln t_0\right)}$ is the representative of $K_p(t_0,\xi )$. Transporting the identity gives the result. □

Proposition 17 

(Proportional modulation). Let ${\xi }_0\in {\mathbb{R}}_{+}$ and define

$$h_p\left(t\right):={\mathrm{Exp}}_p(i_p\odot {\xi }_0\odot t)\odot f_p\left(t\right).$$

Then

$${\mathcal{F}}_p\left\{h_p\right\}\left(\xi \right)={\mathcal{F}}_p\left\{f_p\right\}(\xi \ominus {\xi }_0),\xi >0.$$

Proof. 

The representative of the proportional factor is $e^{i\left(\ln {\xi }_0\right)u}$. Thus, the representative of $h_p$ is $e^{i\left(\ln {\xi }_0\right)u}{f}_c\left(u\right)$. The classical modulation rule gives

$$\mathcal{F}\left\{h_c\right\}\left(\omega \right)=\mathcal{F}\left\{f_c\right\}(\omega -\ln {\xi }_0).$$

Since $\ln (\xi \ominus {\xi }_0)=\ln \xi -\ln {\xi }_0$, the result follows. □

6.3. Scaling

Proposition 18 

(Proportional scaling). Let $a\in {\mathbb{R}}_{+}$, $a\ne 1$, and set $\alpha =\ln a\ne 0$. Here a is a proportional scalar, while α is the classical dilation factor acting on the logarithmic variable. Define $h_p\left(t\right):=f_p(a\odot t)$. Then

$${\mathcal{F}}_p\left\{h_p\right\}\left(\xi \right)={(e\oslash |a|}_p)\odot {\mathcal{F}}_p\left\{f_p\right\}(\xi \oslash a),\xi >0.$$

Proof. 

Let $\alpha =\ln a\ne 0$. The representative of $h_p$ is $h_c\left(u\right)=f_c\left(\alpha u\right)$. The classical scaling rule gives

$$\mathcal{F}\left\{h_c\right\}\left(\omega \right)={\displaystyle \frac{1}{\left|\alpha \right|}}\mathcal{F}\left\{f_c\right\}\left({\displaystyle \frac{\omega }{\alpha }}\right).$$

Since $\ln (\xi \oslash a)=\ln \xi /\ln a$ and $e\oslash {\left|a\right|}_p=\mathcal{E}(1/|\ln a\left|\right)$, the result follows. □

6.4. Transform of Proportional Derivatives

Proposition 19 

(Transform of the proportional derivative). Let $f_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$ be proportionally differentiable, and suppose that its representative $f_c$ satisfies $f_c,f_c^{\prime }\in L^1(\mathbb{R};\mathbb{C})$ and $f_c\left(u\right)\to 0$ as $u\to \pm \infty$. Then

$${\mathcal{F}}_p\left\{D_p{f}_p\right\}\left(\xi \right)=(i_p\odot \xi )\odot {\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right).$$

Proof. 

The representative of $D_p{f}_p$ is $f_c^{\prime }$. Hence, the result follows from the classical identity $\mathcal{F}\left\{f_c^{\prime }\right\}\left(\omega \right)=i\omega \mathcal{F}\left\{f_c\right\}\left(\omega \right)$. □

Corollary 2 

(Higher-order derivatives). Let $n\in \mathbb{N}$. Assume that ${\mathcal{R}}_p{f}_p\in W^{n,1}(\mathbb{R};\mathbb{C})$ and that $D^k\left({\mathcal{R}}_p{f}_p\right)\left(u\right)\to 0$ as $u\to \pm \infty$ for $0\le k\le n-1$. Here $D_p^n{f}_p$ is understood through the representative identity

$$D_p^n{f}_p:={\mathcal{R}}_p^{-1}\left(D^n\left({\mathcal{R}}_p{f}_p\right)\right).$$

If the representative is classically n-times differentiable, this agrees with Definition 13. Then $D_p^n{f}_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$ and

$${\mathcal{F}}_p\left\{D_p^n{f}_p\right\}\left(\xi \right)={(i_p\odot \xi )}^{\left\{n\right\}}\odot {\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right),\xi \in {\mathbb{R}}_{+}.$$

6.5. Proportional Convolution

Definition 17 

(Proportional convolution). Let $f_p,g_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$, and set $f_c={\mathcal{R}}_p{f}_p$, $g_c={\mathcal{R}}_p{g}_p$. Their proportional convolution is the element $f_p{\ast }_p{g}_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$ defined by

$${\mathcal{R}}_p\left(f_p{\ast }_p{g}_p\right)=f_c\ast g_c.$$

Equivalently, whenever the proportional integral is pointwise meaningful, this definition is represented by

$$\left(f_p{\ast }_p{g}_p\right)\left(t\right)={\int }_0^{\infty }{f}_p(t\ominus s)\odot g_p\left(s\right)d_{\odot }s,t>0.$$

Thus, the convolution is first a transported $L^1$-object and only secondarily a pointwise integral formula.

Proposition 20 

(Classical representative of proportional convolution). Let $f_p,g_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$, with representatives $f_c,g_c\in L^1(\mathbb{R};\mathbb{C})$. Then

$${\mathcal{R}}_p\left(f_p{\ast }_p{g}_p\right)=f_c\ast g_c.$$

Proof. 

This is the defining representative identity. If the pointwise integral formula is used, the same identity follows by writing $t=e^u$ and $s=e^v$, so that $t\ominus s=e^{u-v}$ and the representative of the integrand is $f_c(u-v)g_c\left(v\right)$. □

Let

$${\gamma }_p:=\mathcal{E}\left(\sqrt{2\pi }\right)=e^{\sqrt{2\pi }}$$

be the proportional number representing the classical convolution-normalization factor. The following theorem uses the symmetric Fourier normalization fixed in Section 5.1; this is why the factor $\sqrt{2\pi }$ appears.

Theorem 7 

(Convolution theorem). Let $f_p,g_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$. Then

$${\mathcal{F}}_p\left\{f_p{\ast }_p{g}_p\right\}\left(\xi \right)={\gamma }_p\odot {\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)\odot {\mathcal{F}}_p\left\{g_p\right\}\left(\xi \right).$$

The equality holds pointwise for every $\xi >0$, with both sides interpreted through their $C_0(\mathbb{R};\mathbb{C})$-representatives.

Proof. 

This follows from Proposition 20, Theorem 3, and the classical identity $\mathcal{F}(f_c\ast g_c)=\sqrt{2\pi }\mathcal{F}{f}_c\mathcal{F}{g}_c$. □

Proposition 21 

(Transported Young inequality). Let $1\le r\le \infty$, $A_p\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$, and $B_p\in L_p^r({\mathbb{R}}_{+};{\mathbb{C}}_p)$. Define the extended proportional convolution by

$${\mathcal{R}}_p\left(A_p{\ast }_p{B}_p\right)=\left({\mathcal{R}}_p{A}_p\right)\ast \left({\mathcal{R}}_p{B}_p\right).$$

Then $A_p{\ast }_p{B}_p\in L_p^r({\mathbb{R}}_{+};{\mathbb{C}}_p)$ and

$$\parallel A_p{\ast }_p{B}_p{\parallel }_{L_p^r}\le \parallel A_p{\parallel }_{L_p^1}{\parallel B_p\parallel }_{L_p^r}.$$

In particular, the notation $A_p{\ast }_p{B}_p$ is justified for $L_p^1{\ast }_p{L}_p^2$ convolutions used in resolvent formulas.

Proof. 

Apply the classical Young inequality to ${\mathcal{R}}_p{A}_p\in L^1(\mathbb{R};\mathbb{C})$ and ${\mathcal{R}}_p{B}_p\in L^r(\mathbb{R};\mathbb{C})$, and transport the resulting estimate through the representative isometry of Proposition 12. □

These identities show that the proportional Fourier transform preserves the fundamental algebraic and analytic behavior of the classical Fourier transform, but expressed entirely in the language of proportional arithmetic.

7. Proportional Schwartz Space

In this section we introduce the proportional analogue of the classical Schwartz space. The purpose is to identify a natural class of proportional functions on which the proportional Fourier transform is particularly well behaved. Since the proportional Fourier transform is generally ${\mathbb{C}}_p$-valued, the correct invariant space is the complex-valued proportional Schwartz space.

7.1. Definition and Correspondence

The classical complex Schwartz space is

$$\mathcal{S}(\mathbb{R};\mathbb{C})=\left\{f_c\in C^{\infty }(\mathbb{R};\mathbb{C}):\underset{u\in \mathbb{R}}{sup}\left|u^m{f}_c^{\left(n\right)}\left(u\right)\right|<\infty ,\forall m,n\in {\mathbb{N}}_0\right\}.$$

Definition 18 

(Complex proportional Schwartz space). The complex proportional Schwartz space is defined intrinsically as the transported Fréchet space

$${\mathcal{S}}_p({\mathbb{R}}_{+};{\mathbb{C}}_p):={\mathcal{R}}_p^{-1}\left(\mathcal{S}(\mathbb{R};\mathbb{C})\right).$$

Its topology is generated by the ordinary real seminorms

$$p_{m,n}^{\left(p\right)}\left(f_p\right):=\underset{u\in \mathbb{R}}{sup}\left|u^m\left(D^n{\mathcal{R}}_p{f}_p\right)\left(u\right)\right|,m,n\in {\mathbb{N}}_0.$$

Equivalently, writing $u=\ln x$, membership means that $f_p$ is infinitely proportionally differentiable and that the proportional expression

$${\left|x^{\left\{m\right\}}\odot D_p^n{f}_p\left(x\right)\right|}_p$$

has bounded representative for every $m,n\in {\mathbb{N}}_0$. The seminorms, norms, and topological boundedness conditions are ordinary nonnegative real quantities; the proportional modulus itself is a value in the embedded proportional field. The real proportional Schwartz space is the subspace with real-valued representative.

Theorem 8 

(Schwartz correspondence). Let $f_c\in C^{\infty }(\mathbb{R};\mathbb{C})$ and define $f_p\left(x\right)=\mathcal{E}\left(f_c\left(\ln x\right)\right)$. Then

$$f_c\in \mathcal{S}(\mathbb{R};\mathbb{C})⟺f_p\in {\mathcal{S}}_p({\mathbb{R}}_{+};{\mathbb{C}}_p).$$

Proof. 

For $m,n\in {\mathbb{N}}_0$,

$$D_p^n{f}_p\left(x\right)=\mathcal{E}\left(f_c^{\left(n\right)}\left(\ln x\right)\right),x^{\left\{m\right\}}=\mathcal{E}\left({\left(\ln x\right)}^m\right).$$

Thus,

$$x^{\left\{m\right\}}\odot D_p^n{f}_p\left(x\right)=\mathcal{E}\left({\left(\ln x\right)}^m{f}_c^{\left(n\right)}\left(\ln x\right)\right).$$

Taking the proportional modulus gives

$${\left|x^{\left\{m\right\}}\odot D_p^n{f}_p\left(x\right)\right|}_p=\mathcal{E}\left(\left|{\left(\ln x\right)}^m{f}_c^{\left(n\right)}\left(\ln x\right)\right|\right).$$

Since $u=\ln x$ ranges over all $\mathbb{R}$, the proportional boundedness condition is equivalent to the classical Schwartz boundedness condition. □

7.2. Stability Under the Proportional Fourier Transform

Theorem 9 

(Invariance of ${\mathcal{S}}_p({\mathbb{R}}_{+};{\mathbb{C}}_p)$). The proportional Fourier transform is a continuous linear automorphism of the Fréchet space ${\mathcal{S}}_p({\mathbb{R}}_{+};{\mathbb{C}}_p)$; its inverse ${\mathcal{F}}_p^{-1}$ is also continuous.

Proof. 

If $f_p\in {\mathcal{S}}_p({\mathbb{R}}_{+};{\mathbb{C}}_p)$, then ${\mathcal{R}}_p{f}_p\in \mathcal{S}(\mathbb{R};\mathbb{C})$ by Theorem 8. Since the classical Fourier transform maps $\mathcal{S}(\mathbb{R};\mathbb{C})$ continuously and bijectively onto itself, $\mathcal{F}\left\{{\mathcal{R}}_p{f}_p\right\}\in \mathcal{S}(\mathbb{R};\mathbb{C})$. By Theorem 3, ${\mathcal{R}}_p\left({\mathcal{F}}_p\left\{f_p\right\}\right)=\mathcal{F}\left\{{\mathcal{R}}_p{f}_p\right\}$, and another application of Theorem 8 gives ${\mathcal{F}}_p\left\{f_p\right\}\in {\mathcal{S}}_p({\mathbb{R}}_{+};{\mathbb{C}}_p)$. The inverse proportional transform corresponds to the classical inverse Fourier transform under ${\mathcal{R}}_p$, so the same argument applied to ${\mathcal{F}}^{-1}$ gives surjectivity. Because ${\mathcal{R}}_p$ is a homeomorphism for the transported seminorms and both $\mathcal{F}$ and ${\mathcal{F}}^{-1}$ are continuous on $\mathcal{S}(\mathbb{R};\mathbb{C})$, both ${\mathcal{F}}_p$ and ${\mathcal{F}}_p^{-1}$ are continuous. Hence, ${\mathcal{F}}_p$ is a continuous linear automorphism. □

Theorem 10 

(Fourier inversion on ${\mathcal{S}}_p({\mathbb{R}}_{+};{\mathbb{C}}_p)$). For every $f_p\in {\mathcal{S}}_p({\mathbb{R}}_{+};{\mathbb{C}}_p)$,

$${\mathcal{F}}_p^{-1}\left\{{\mathcal{F}}_p\left\{f_p\right\}\right\}=f_p.$$

Proof. 

By Theorem 8, ${\mathcal{R}}_p{f}_p\in \mathcal{S}(\mathbb{R};\mathbb{C})$. Classical Fourier inversion holds pointwise on $\mathcal{S}(\mathbb{R};\mathbb{C})$, and Theorem 3 identifies the representative of ${\mathcal{F}}_p\left\{f_p\right\}$ with $\mathcal{F}\left\{{\mathcal{R}}_p{f}_p\right\}$. Applying ${\mathcal{R}}_p^{-1}$ gives ${\mathcal{F}}_p^{-1}\left\{{\mathcal{F}}_p\left\{f_p\right\}\right\}=f_p$ on ${\mathbb{R}}_{+}$. □

Remark 11. 

If $f_p$ is ${\mathbb{R}}_{+}$-valued, ${\mathcal{F}}_p\left\{f_p\right\}$ need not be ${\mathbb{R}}_{+}$-valued. This is why the invariant Schwartz space is ${\mathcal{S}}_p({\mathbb{R}}_{+};{\mathbb{C}}_p)$, not only its real subspace.

7.3. Examples

Example 2 

(Proportional Gaussian). Let $a>0$ and $f_c\left(u\right)=e^{-a^2{u}^2}$. Since $f_c\in \mathcal{S}\left(\mathbb{R}\right)$, the proportional representative

$$f_p\left(x\right)=\exp \left(e^{-a^2{\left(\ln x\right)}^2}\right)$$

belongs to ${\mathcal{S}}_p\left({\mathbb{R}}_{+}\right)$. Although $f_p\left(x\right)\to 1$ as $x\to 0^{+}$ or $x\to \infty$, this is decay toward the proportional zero $1=\mathcal{E}\left(0\right)$, because the representative tends to the classical zero. With the symmetric normalization,

$$\mathcal{F}\left\{e^{-a^2{u}^2}\right\}\left(\omega \right)={\displaystyle \frac{1}{a\sqrt{2}}}\exp \left(-{\displaystyle \frac{{\omega }^2}{4a^2}}\right).$$

Therefore,

$${\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)=\mathcal{E}\left({\displaystyle \frac{1}{a\sqrt{2}}}\exp \left(-{\displaystyle \frac{{\left(\ln \xi \right)}^2}{4a^2}}\right)\right).$$

In this case, the proportional transform remains real because its representative is real and even.

Figure 1 compares the centered Gaussian representative with its proportional realization on linear and logarithmic x-scales. It illustrates the passage from additive symmetry in u to multiplicative symmetry about $x=1$.

Example 3 

(A genuinely complex proportional transform pair). Let $f_c\left(u\right)=u{e}^{-u^2}$ and define $f_p\left(x\right)=\mathcal{E}\left(\left(\ln x\right)e^{-{\left(\ln x\right)}^2}\right)$. Then $f_p\in {\mathcal{S}}_p\left({\mathbb{R}}_{+}\right)$, but its proportional Fourier transform is genuinely ${\mathbb{C}}_p$-valued. Indeed,

$$\mathcal{F}\left\{u{e}^{-u^2}\right\}\left(\omega \right)=-{\displaystyle \frac{i\omega }{2\sqrt{2}}}{e}^{-{\omega }^2/4},$$

and therefore,

$${\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)=\mathcal{E}\left(-{\displaystyle \frac{i\ln \xi }{2\sqrt{2}}}{e}^{-{\left(\ln \xi \right)}^2/4}\right).$$

This example shows why complex proportional function spaces are necessary even when the input representative is real-valued.

Example 4 

(Exponential decay). Let $a>0$ and $f_c\left(u\right)=e^{-a\left|u\right|}$. Its proportional representative is

$$f_p\left(x\right)=\exp \left(e^{-a\left|\ln x\right|}\right).$$

Although $f_p$ does not belong to ${\mathcal{S}}_p\left({\mathbb{R}}_{+}\right)$, it belongs to $L_p^1\left({\mathbb{R}}_{+}\right)$. Since

$$\mathcal{F}\left\{e^{-a\left|u\right|}\right\}\left(\omega \right)=\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle \frac{a}{a^2+{\omega }^2}},$$

we obtain

$${\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)=\mathcal{E}\left(\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle \frac{a}{a^2+{\left(\ln \xi \right)}^2}}\right).$$

The proportional Schwartz space provides a natural domain for the proportional Fourier transform. On this space, the transform is invertible, stable and compatible with proportional differentiation and multiplication by proportional powers.

8. Proportional Sobolev Spaces

In this section we introduce proportional Sobolev spaces. These spaces measure proportional regularity through the representative map while preserving the proportional notation in the original variable. The classical Sobolev and Fourier-characterization results used here are standard; see, for example, [16,23]. As in the Schwartz case, the complex-valued version is the natural one for Fourier analysis.

Definition 19 

(Complex proportional Sobolev space). Let $s\ge 0$. The proportional Sobolev space $H_p^s({\mathbb{R}}_{+};{\mathbb{C}}_p)$ is

$$H_p^s({\mathbb{R}}_{+};{\mathbb{C}}_p):=\{f_p:{\mathbb{R}}_{+}\to {\mathbb{C}}_p:{\mathcal{R}}_p{f}_p\in H^s(\mathbb{R};\mathbb{C})\},$$

with the Fourier-side norm

$$\parallel f_p{\parallel }_{H_p^s}^2:={\int }_{\mathbb{R}}{(1+|\omega |}^2{)}^s{\left|\mathcal{F}\left\{{\mathcal{R}}_p{f}_p\right\}\left(\omega \right)\right|}^{2}d\omega .$$

This equality is the definition of the norm used in the paper, with the symmetric Fourier normalization. The derivative-sum norm for integer order is only equivalent to this norm. The real proportional Sobolev space $H_p^s\left({\mathbb{R}}_{+}\right)$ is the subspace whose representatives are real-valued.

Remark 12 

(Scope of the Sobolev scale). This paper works with $s\ge 0$. Negative-order proportional Sobolev spaces can be defined by duality or by the Fourier representative, but their systematic treatment requires proportional tempered distributions and is left for future work.

Theorem 11 

(Sobolev correspondence). The representative map

$${\mathcal{R}}_p:H_p^s({\mathbb{R}}_{+};{\mathbb{C}}_p)\to H^s(\mathbb{R};\mathbb{C}),{\mathcal{R}}_p\left(f_p\right)\left(u\right)={\mathsf{\ell }}_p\left(f_p\left(e^u\right)\right),$$

is an isometric isomorphism. Its inverse is

$${\mathcal{R}}_p^{-1}\left(f_c\right)\left(x\right)=\mathcal{E}\left(f_c\left(\ln x\right)\right).$$

Proof. 

This follows directly from the definition of $H_p^s({\mathbb{R}}_{+};{\mathbb{C}}_p)$ and the transported norm. □

Theorem 12 

(Fourier characterization of $H_p^s({\mathbb{R}}_{+};{\mathbb{C}}_p)$). Let $s\ge 0$. In this theorem ${\mathcal{F}}_p$ is understood as the unitary $L_p^2$-extension of Theorem 6; if $f_p\in L_p^1\cap H_p^s$, it agrees with the integral transform of Definition 14. A proportional function $f_p$ belongs to $H_p^s({\mathbb{R}}_{+};{\mathbb{C}}_p)$ if and only if

$${\int }_{\mathbb{R}}{(1+|\omega |}^2{)}^s{\left|{\mathcal{R}}_p\left({\mathcal{F}}_p\left\{f_p\right\}\right)\left(\omega \right)\right|}^{2}d\omega <\infty .$$

Equivalently, using the correspondence theorem,

$${\int }_{\mathbb{R}}{(1+|\omega |}^2{)}^s{\left|\mathcal{F}\left\{{\mathcal{R}}_p{f}_p\right\}\left(\omega \right)\right|}^{2}d\omega <\infty .$$

Proof. 

This is the Fourier characterization of $H^s(\mathbb{R};\mathbb{C})$, transported through Theorem 3. □

Definition 20 

(Weak proportional derivatives). Let $f_p\in L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$ and let $k\in \mathbb{N}$. We say that the weak proportional derivative $D_{p,w}^k{f}_p$ exists in $L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$ if the distributional derivative $D^k\left({\mathcal{R}}_p{f}_p\right)$ is represented by an $L^2(\mathbb{R};\mathbb{C})$ function. In that case we define

$$D_{p,w}^k{f}_p:={\mathcal{R}}_p^{-1}\left(D^k\left({\mathcal{R}}_p{f}_p\right)\right).$$

We also set $D_{p,w}^0{f}_p=f_p$. If ${\mathcal{R}}_p{f}_p$ is classically k-times differentiable with suitable integrability, this weak derivative agrees with the proportional derivative of Definition 13.

Remark 13 

(Intrinsic distributional formulation with Haar measure). The weak proportional derivative can equivalently be characterized directly on $(0,\infty )$ with Haar measure $dx/x$. Let ${\mathcal{D}}_p:={\mathcal{R}}_p^{-1}\left(C_c^{\infty }(\mathbb{R};\mathbb{C})\right)$. A function $g_p\in L_p^2$ is the first weak proportional derivative of $f_p$ if and only if, for every ${\psi }_p\in {\mathcal{D}}_p$,

$${\int }_0^{\infty }{\mathsf{\ell }}_p\left(g_p\left(x\right)\right)\overline{{\mathsf{\ell }}_p\left({\psi }_p\left(x\right)\right)}{\displaystyle \frac{dx}{x}}=-{\int }_0^{\infty }{\mathsf{\ell }}_p\left(f_p\left(x\right)\right)\overline{{\displaystyle \frac{d}{d\ln x}}{\mathsf{\ell }}_p\left({\psi }_p\left(x\right)\right)}{\displaystyle \frac{dx}{x}}.$$

The conjugation bars are essential because the paper uses the standard sesquilinear complex Hilbert-space convention. This is an intrinsic half-line formulation of the same transported distributional identity; the operator $d/d\ln x=xd/dx$ is the infinitesimal generator of multiplicative translations.

Theorem 13 

(Weak derivative characterization for integer order). Let $m\in \mathbb{N}$. A proportional function $f_p\in L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$ belongs to $H_p^m({\mathbb{R}}_{+};{\mathbb{C}}_p)$ if and only if

$$D_{p,w}^k{f}_p\in L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p),k=0,1,\dots ,m.$$

Moreover,

$$\parallel f_p{\parallel }_{H_p^m}^2\asymp {\displaystyle \sum _{k=0}^m}{\parallel D_{p,w}^k{f}_p\parallel }_{L_p^2}^2.$$

Here the equivalence constants depend only on m.

Proof. 

By Definition 20, the representative of $D_{p,w}^k{f}_p$ is the distributional derivative $D^k\left({\mathcal{R}}_p{f}_p\right)$. Hence, the statement is exactly the classical weak-derivative characterization of $H^m(\mathbb{R};\mathbb{C})$, transported through ${\mathcal{R}}_p$. □

Proposition 22. 

For every $s\ge 0$,

$${\mathcal{S}}_p({\mathbb{R}}_{+};{\mathbb{C}}_p)\subset H_p^s({\mathbb{R}}_{+};{\mathbb{C}}_p).$$

Proof. 

Let $f_p\in {\mathcal{S}}_p({\mathbb{R}}_{+};{\mathbb{C}}_p)$. By Theorem 8, ${\mathcal{R}}_p{f}_p\in \mathcal{S}(\mathbb{R};\mathbb{C})$. The classical embedding $\mathcal{S}(\mathbb{R};\mathbb{C})\subset H^s(\mathbb{R};\mathbb{C})$ holds for every $s\ge 0$. Therefore, ${\mathcal{R}}_p{f}_p\in H^s(\mathbb{R};\mathbb{C})$, and the Sobolev correspondence of Theorem 11 gives $f_p\in H_p^s({\mathbb{R}}_{+};{\mathbb{C}}_p)$. □

Example 5 

(Sobolev regularity of a proportional exponential profile). Let $a>0$ and $f_c\left(u\right)=e^{-a\left|u\right|}$. Then

$$f_p\left(x\right)=\exp \left(e^{-a\left|\ln x\right|}\right).$$

Since

$$\mathcal{F}\left\{f_c\right\}\left(\omega \right)=\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle \frac{a}{a^2+{\omega }^2}},$$

we have $f_p\in H_p^s\left({\mathbb{R}}_{+}\right)$ if and only if

$${\int }_{\mathbb{R}}{(1+|\omega |}^2{)}^s{\left|\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle \frac{a}{a^2+{\omega }^2}}\right|}^{2}d\omega <\infty .$$

For large $\left|\omega \right|$, the integrand behaves like ${\left|\omega \right|}^{2s-4}$. Hence, the integral converges if and only if $s<3/2$.

9. Applications to Proportional Differential Equations

In this section we illustrate how the proportional Fourier transform can be used to study proportional differential equations. The guiding principle is that proportional differentiation becomes proportional multiplication in the proportional frequency variable. Consequently, equations written with $D_p$ can often be transformed into proportional algebraic equations and then returned to the original proportional variable by ${\mathcal{F}}_p^{-1}$.

9.1. A Proportional Gaussian Equation

Let $a>0$ and consider $f_c\left(u\right)=e^{-a^2{u}^2}$. It satisfies

$$f_c^{\prime }\left(u\right)+2a^{2}u{f}_c\left(u\right)=0.$$

The proportional representative

$$f_p\left(x\right)=\exp \left(e^{-a^2{\left(\ln x\right)}^2}\right)$$

satisfies

$$D_p{f}_p\left(x\right)\oplus \left(e^{2a^2}\odot x\odot f_p\left(x\right)\right)=1.$$

(19)

Indeed, the representative of the left-hand side is $f_c^{\prime }\left(\ln x\right)+2a^2\left(\ln x\right)f_c\left(\ln x\right)$, which is identically zero. Since 1 is the proportional zero, (19) follows.

Applying the proportional Fourier transform gives

$${\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)=\mathcal{E}\left({\displaystyle \frac{1}{a\sqrt{2}}}\exp \left(-{\displaystyle \frac{{\left(\ln \xi \right)}^2}{4a^2}}\right)\right).$$

This example shows that the proportional Gaussian is not only a proportional Schwartz function, but also solves a proportional differential equation naturally associated with its classical representative.

9.2. A Proportional Resolvent Equation

Let $\alpha >0$ and let $F\in L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$. We seek $U\in H_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$ satisfying the following proportional equation in the transported weak sense:

$$(1\ominus D_{p,w}^{2}U)\oplus (e^{{\alpha }^2}\odot U)=F.$$

(20)

For smooth representatives, $D_{p,w}^{2}U$ coincides with the second proportional derivative $D_p^{2}U$; in $H_p^2$, it is understood as the weak proportional derivative of Definition 20. If $U=\mathcal{E}(v\circ \ln )$ and $F=\mathcal{E}(f\circ \ln )$, then (20) has representative

$$-v^{\prime \prime }\left(X\right)+{\alpha }^{2}v\left(X\right)=f\left(X\right).$$

Thus, (20) is a proportional formulation of the classical resolvent problem transported through the proportional arithmetic.

Theorem 14 

(Solution of the proportional resolvent equation). Let $\alpha >0$ and $F\in L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$. Then (20) has a unique solution $U\in H_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$, given equivalently by

$$U=G_{\alpha }{\ast }_{p}F={\mathcal{F}}_p^{-1}\left\{M_{\alpha ,p}\odot {\mathcal{F}}_p\left\{F\right\}\right\},M_{\alpha ,p}\left(\xi \right):=\mathcal{E}\left({\displaystyle \frac{1}{{\left(\ln \xi \right)}^2+{\alpha }^2}}\right),$$

where

$$G_{\alpha }\left(x\right)=\mathcal{E}\left({\displaystyle \frac{1}{2\alpha }}{e}^{-\alpha \left|\ln x\right|}\right).$$

Here $G_{\alpha }\in L_p^1({\mathbb{R}}_{+};{\mathbb{C}}_p)$, and the convolution $G_{\alpha }{\ast }_{p}F$ is understood in the transported $L_p^1{\ast }_p{L}_p^2$ sense of Proposition 21. Moreover,

$${\parallel U\parallel }_{H_p^2}\le C_{\alpha }{\parallel F\parallel }_{L_p^2}$$

for a constant $C_{\alpha }>0$ depending only on α.

Proof. 

Let $f={\mathcal{R}}_{p}F$. Taking the classical Fourier transform of the representative equation $-v^{\prime \prime }+{\alpha }^{2}v=f$ gives

$$({\omega }^2+{\alpha }^2)\widehat{v}\left(\omega \right)=\widehat{f}\left(\omega \right),\widehat{v}\left(\omega \right)={\displaystyle \frac{\widehat{f}\left(\omega \right)}{{\omega }^2+{\alpha }^2}}.$$

The multiplier ${({\omega }^2+{\alpha }^2)}^{-1}$ defines a unique $v\in H^2(\mathbb{R};\mathbb{C})$ for each $f\in L^2(\mathbb{R};\mathbb{C})$. Moreover,

$${\parallel v\parallel }_{H^2}^2={\int }_{\mathbb{R}}{(1+{\omega }^2)}^2{\displaystyle \frac{|\widehat{f}{\left(\omega \right)|}^2}{{({\omega }^2+{\alpha }^2)}^2}}d\omega \le C_{\alpha }^2{\parallel f\parallel }_{L^2}^2,$$

where $C_{\alpha }={sup}_{\omega \in \mathbb{R}}(1+{\omega }^2)/({\omega }^2+{\alpha }^2)<\infty$. With the symmetric Fourier normalization, one has

$$\mathcal{F}\left\{{\displaystyle \frac{1}{2\alpha }}{e}^{-\alpha |·|}\right\}\left(\omega \right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\displaystyle \frac{1}{{\omega }^2+{\alpha }^2}}.$$

Therefore, $g_{\alpha }\left(X\right)={\left(2\alpha \right)}^{-1}{e}^{-\alpha \left|X\right|}$ is the convolution kernel associated with the multiplier $m_{\alpha }\left(\omega \right)={({\omega }^2+{\alpha }^2)}^{-1}$, because

$$\mathcal{F}(g_{\alpha }\ast f)\left(\omega \right)=\sqrt{2\pi }{\widehat{g}}_{\alpha }\left(\omega \right)\widehat{f}\left(\omega \right)={\displaystyle \frac{\widehat{f}\left(\omega \right)}{{\omega }^2+{\alpha }^2}}.$$

Thus, $g_{\alpha }\in L^1\left(\mathbb{R}\right)$ and $v=g_{\alpha }\ast f$, with the convolution understood as the standard $L^1\ast L^2\to L^2$ convolution. Transporting these identities through ${\mathcal{R}}_p^{-1}$ and Proposition 21 gives $U=G_{\alpha }{\ast }_{p}F$. The same Fourier-domain identity gives the multiplier form with $M_{\alpha ,p}$. Uniqueness in $H_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$ and the stated estimate follow from the uniqueness and estimate for the representative problem. □

9.3. A Proportional Heat Equation

Definition 21 

(Partial proportional derivatives). Let $U:{\mathbb{R}}_{+}\times [1,\infty )\to {\mathbb{C}}_p$ and let $W(X,T)={\mathsf{\ell }}_p\left(U(e^X,e^T)\right)$ be its representative, where $X=\ln x$ and $T=\ln \tau$. Whenever the corresponding representative derivatives exist, define

$$D_{p,x}U(x,\tau ):=\mathcal{E}\left(W_X(\ln x,\ln \tau )\right),D_{p,\tau }U(x,\tau ):=\mathcal{E}\left(W_T(\ln x,\ln \tau )\right),$$

and

$$D_{p,x}^{2}U(x,\tau ):=\mathcal{E}\left(W_{XX}(\ln x,\ln \tau )\right).$$

Weak partial proportional derivatives are defined analogously by replacing the representative derivatives by distributional derivatives. This definition keeps the partial differential equation in proportional notation while making the transported classical meaning precise.

Let $U:{\mathbb{R}}_{+}\times [1,\infty )\to {\mathbb{C}}_p$ and write

$$U(x,\tau )=\mathcal{E}\left(W\right(\ln x,\ln \tau \left)\right).$$

The letter W is used for the classical representative in this subsection in order to avoid confusing the proportional unknown U with its representative. Let $X=\ln x$ and $T=\ln \tau$. Here $e^{\kappa }=\mathcal{E}\left(\kappa \right)$ denotes the proportional scalar corresponding to the classical coefficient $\kappa$, not an ordinary exponential coefficient. The proportional heat equation is

$$D_{p,\tau }U=e^{\kappa }\odot D_{p,x}^{2}U,\kappa >0,$$

(21)

with initial condition $U(x,1)=\Phi \left(x\right)$. Its representative is the classical heat equation

$$W_T=\kappa W_{XX},W(X,0)=\phi \left(X\right),$$

where $\Phi \left(x\right)=\mathcal{E}\left(\phi \right(\ln x\left)\right)$. For general $L_p^2$ data, the displayed equation is the proportional notation for the transported semigroup problem; pointwise proportional derivatives require additional regularity.

Definition 22 

(Mild proportional heat solution). A function $U\in C([1,\infty );L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p))$ is a mild solution of (21) with initial datum $\Phi \in L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$ if its representative $W(X,T)={\mathcal{R}}_p\left(U(·,e^T)\right)\left(X\right)$ is the classical mild solution of $W_T=\kappa W_{XX}$ with initial datum $\phi ={\mathcal{R}}_p\Phi$.

Theorem 15 

(Solution of the proportional heat equation). Let $\kappa >0$ and $\Phi \in L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$. Then the proportional heat problem

$$\left\{\begin{array}{cc}{D}_{p,\tau }U=e^{\kappa }\odot D_{p,x}^{2}U,\hfill & x>0,\tau >1,\hfill \\ U(x,1)=\Phi \left(x\right),\hfill & x>0,\hfill \end{array}\right.$$

has a unique mild solution $U\in C([1,\infty );L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p))$. It is given, in $L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$, by

$$U(·,\tau )={\mathcal{F}}_p^{-1}\left\{H_p(\xi ,\tau )\odot {\mathcal{F}}_p\{\Phi \}\left(\xi \right)\right\}.$$

If the initial datum has additional regularity and integrability, the same formula admits the pointwise notation

$$U(x,\tau )={\mathcal{F}}_p^{-1}\left\{H_p(\xi ,\tau )\odot {\mathcal{F}}_p\{\Phi \}\left(\xi \right)\right\}\left(x\right),$$

where

$$H_p(\xi ,\tau )=\mathcal{E}\left(e^{-\kappa {\left(\ln \xi \right)}^2\ln \tau }\right).$$

Moreover,

$${\parallel U(·,\tau )\parallel }_{L_p^2}\le {\parallel \Phi \parallel }_{L_p^2},\tau \ge 1.$$

For general $\Phi \in L_p^2$, Equation (21) is satisfied in the transported mild semigroup sense. A pointwise or strong interpretation requires additional regularity on Φ. If Φ has a real-valued representative, then the mild solution also has a real-valued representative; hence, it remains ${\mathbb{R}}_{+}$-valued.

Proof. 

The representative problem is the classical heat equation on $\mathbb{R}$. Its mild solution is $W\left(T\right)=e^{\kappa T{\partial }_X^2}\phi$, and in Fourier variables $\widehat{W}(\omega ,T)=e^{-\kappa {\omega }^{2}T}\widehat{\phi }\left(\omega \right)$. Transporting this formula through ${\mathcal{F}}_p$ gives the stated proportional solution. The estimate follows from the contraction property of the classical heat semigroup in $L^2(\mathbb{R};\mathbb{C})$ [24,25,26]. If $\phi$ is real-valued, the heat semigroup preserves real-valuedness, which proves the final assertion. □

Corollary 3 

(Semigroup law in proportional time). Let ${\mathcal{T}}_p\left(\tau \right)\Phi :=U(·,\tau )$ denote the mild solution operator for $\tau \ge 1$. Then

$${\mathcal{T}}_p\left(1\right)=\mathrm{Id},{\mathcal{T}}_p({\tau }_1\oplus {\tau }_2)={\mathcal{T}}_p\left({\tau }_1\right){\mathcal{T}}_p\left({\tau }_2\right),{\tau }_1,{\tau }_2\ge 1,$$

where ${\tau }_1\oplus {\tau }_2={\tau }_1{\tau }_2$. Thus, ordinary addition of representative times $T_j=\ln {\tau }_j$ becomes proportional addition of the positive time variables.

Proof. 

Under representatives, ${\mathcal{T}}_p\left(\tau \right)$ is the classical heat semigroup at time $T=\ln \tau$. Hence,

$$e^{(T_1+T_2)\kappa {\partial }_X^2}=e^{T_1\kappa {\partial }_X^2}{e}^{T_2\kappa {\partial }_X^2},$$

and $T_1+T_2=\ln \left({\tau }_1{\tau }_2\right)=\ln ({\tau }_1\oplus {\tau }_2)$. □

Remark 14 

(Strong proportional heat solutions). If ${\mathcal{R}}_p\Phi \in H^2(\mathbb{R};\mathbb{C})$, then the mild solution is strong for $T>0$ and satisfies the representative heat equation in $L^2(\mathbb{R};\mathbb{C})$. Transported back to the proportional setting, U satisfies (21) in $L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$. Higher regularity of the initial representative gives the corresponding higher-order strong proportional interpretation.

Corollary 4 

(Proportional heat regularization). Let $s\ge 0$, $\tau >1$, and let U be the mild solution of Theorem 15. Then $U(·,\tau )\in H_p^s({\mathbb{R}}_{+};{\mathbb{C}}_p)$, and there exists a constant $C_s>0$, depending only on s, such that

$${\parallel U(·,\tau )\parallel }_{H_p^s}\le C_s\left(1+{\left(\kappa \ln \tau \right)}^{-s/2}\right){\parallel \Phi \parallel }_{L_p^2},\tau >1.$$

Thus, positive proportional heat evolution regularizes the initial datum in the transported Sobolev scale for every positive proportional time $\tau >1$.

Proof. 

In the representative variable $T=\ln \tau >0$, the Fourier multiplier of the heat semigroup is $e^{-\kappa T{\omega }^2}$. Hence,

$${\parallel U(·,\tau )\parallel }_{H_p^s}^2={\int }_{\mathbb{R}}{(1+{\omega }^2)}^s{e}^{-2\kappa T{\omega }^2}|\widehat{\phi }\left(\omega \right){|}^{2}d\omega \le \underset{\omega \in \mathbb{R}}{sup}{(1+{\omega }^2)}^s{e}^{-2\kappa T{\omega }^2}{\parallel \Phi \parallel }_{L_p^2}^2.$$

The elementary bound ${sup}_{r\ge 0}{(1+r)}^s{e}^{-2\kappa Tr}\le C_s^2(1+{\left(\kappa T\right)}^{-s})$ gives the stated estimate after taking square roots and using $T=\ln \tau$. The constant $C_s$ is tied to the symmetric Fourier normalization and the Fourier-side Sobolev norm fixed in Definition 19. □

Remark 15 

(Behavior near the initial proportional time). As $\tau \downarrow 1$, one has $\ln \tau \downarrow 0$, so the displayed $L_p^2\to H_p^s$ smoothing bound deteriorates. This does not indicate blow-up of the mild solution: strong continuity of the classical heat semigroup gives $U(·,\tau )\to \Phi$ in $L_p^2$ as $\tau \downarrow 1$. Rather, the deterioration records that no uniform positive-order smoothing estimate can hold at the initial proportional time for arbitrary $L_p^2$ data.

The examples above show how proportional differential equations can be treated using the proportional Fourier transform. They are model applications illustrating the proportional Fourier calculus: their role is to demonstrate that equations posed natively in proportional arithmetic can be solved coherently within the proportional framework. The method follows the pattern

$$\begin{array}{c}\mathrm{proportional}\mathrm{differential}\mathrm{equation}\hfill \\ ⟶\mathrm{proportional}\mathrm{algebraic}\mathrm{equation}\mathrm{in}\mathrm{frequency}\hfill \\ ⟶\mathrm{inverse}\mathrm{proportional}\mathrm{transform}.\hfill \end{array}$$

9.4. A Scale-Localized Multiplicative Profile

A setting in which the proportional variables have a direct interpretation is a positive response localized around a preferred multiplicative scale. Let $A>0$, $\sigma >0$, and $t_0>0$, and define

$$f_c\left(u\right)=A\exp \left(-{\displaystyle \frac{{(u-\ln t_0)}^2}{2{\sigma }^2}}\right),f_p\left(t\right)=\mathcal{E}\left(f_c\left(\ln t\right)\right).$$

The representative is Gaussian in log-scale, so equal ratios $t/t_0$ and $t_0/t$ produce equal responses. This symmetry is natural for particle sizes, frequencies, concentrations, and other positive quantities measured by fold changes rather than additive offsets. Direct calculation gives

$${\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)=\mathcal{E}\left(A\sigma \exp \left[-{\displaystyle \frac{{\sigma }^2{\left(\ln \xi \right)}^2}{2}}\right]e^{-i\left(\ln t_0\right)\left(\ln \xi \right)}\right).$$

Proposition 23 

(Scale-location and bandwidth decomposition). For the scale-localized family above,

$$\left|{\mathsf{\ell }}_p\left({\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)\right)\right|=A\sigma \exp \left[-{\displaystyle \frac{{\sigma }^2{\left(\ln \xi \right)}^2}{2}}\right]$$

is independent of $t_0$, whereas

$$arg{\mathsf{\ell }}_p\left({\mathcal{F}}_p\left\{f_p\right\}\left(\xi \right)\right)=-\left(\ln t_0\right)\left(\ln \xi \right)\left(\mathrm{mod}2\pi \right).$$

Hence, proportional translation of the preferred scale changes only the spectral phase, while the log-scale width determines the spectral modulus.

Moreover, $f_p\left(t\right)\to 1=\mathcal{E}\left(0\right)$ as $t\to 0^{+}$ and as $t\to \infty$, so the profile approaches the proportional zero at both boundaries. Thus, the spectral modulus determines the log-scale width $\sigma$, while the phase records the preferred scale $t_0$. The proportional formulation keeps the input, output, neutral value, and scale ratios in their original positive domains; the representative formula supplies exact computation and verification. Figure 2 numerically visualizes the profile for $A=1$, $\sigma =0.65$, and $t_0=4$. This example demonstrates an intrinsic scale-location invariant: changing the preferred multiplicative scale leaves the spectral modulus unchanged and modifies only the phase. It provides interpretive usefulness in the positive variable, but no computational advantage over the logarithmic representative is asserted.

10. Discussion: Relation with Classical Fourier Analysis and Mellin-Type Structures

The proportional Fourier transform developed in this paper is structurally connected with the representative map induced by proportional arithmetic. This connection is unavoidable and useful: it explains why the proportional transform inherits inversion, Riemann–Lebesgue, Plancherel, convolution and Sobolev-regularity properties from the classical Fourier transform. At the same time, the representative map is not the language in which the transform is formulated. The proportional construction begins with the operations $\oplus ,\ominus ,\odot ,\oslash$, then defines the proportional derivative, proportional integral, proportional complex field and proportional exponential kernel, and only then obtains the transform ${\mathcal{F}}_p$.

This point is important for positioning the contribution. The paper does not claim that ${\mathcal{F}}_p$ contains analytic information inaccessible to classical Fourier analysis. Instead, it gives an internally consistent proportional-arithmetic realization of harmonic analysis. The representative theorem is a rigorous bridge: it permits the use of classical results while preserving the proportional notation and the proportional algebra in the original problem. This is particularly relevant when the natural variables of a model are positive and the operations of interest are ratios, proportional changes and multiplicative interactions.

What remains genuinely proportional is therefore not a hidden analytic theorem beyond classical Fourier analysis, but the full internal formulation: the variables are positive-scale variables, the zero and unit are 1 and e, algebraic operations are $\oplus ,\ominus ,\odot ,\oslash$, oscillations are encoded by the proportional kernel $K_p$, and regularity is measured in proportional function spaces. The representative map verifies these statements, while the proportional notation records the arithmetic in which the original problem is posed.

There is a close relation with Fourier analysis on the multiplicative group $(0,\infty )$ and with Mellin-type transforms. Under $u=\ln t$, the Haar measure $dt/t$ on $(0,\infty )$ becomes $du$, and multiplicative scaling becomes additive translation. This is the same structural mechanism that appears in Mellin analysis and abstract harmonic analysis on locally compact abelian groups [17,18,19,21,27]. However, the proportional Fourier transform is not introduced as the ordinary Mellin transform of $f_p$. It is the Fourier transform of the proportional representative ${\mathcal{R}}_p{f}_p$, transported back to ${\mathbb{C}}_p$. Thus, the analytic bridge is classical, while the algebraic formulation remains proportional.

The construction is also related to earlier Fourier-type transforms developed in other non-Diophantine arithmetics, such as Cantor-set arithmetic frameworks [22]. This comparison is important: it shows that the present paper is not the first instance of Fourier analysis in a transported arithmetic. Its narrower contribution is the systematic proportional-arithmetic realization on ${\mathbb{R}}_{+}$, including the formal proportional complex field, proportional Schwartz and Sobolev spaces, Plancherel theory and proportional differential- equation examples.

The complex-valued formulation is also essential. Even if a proportional input has a real-valued representative, its Fourier transform may have a complex-valued representative. Consequently, $L_p^q({\mathbb{R}}_{+};{\mathbb{C}}_p)$, ${\mathcal{S}}_p({\mathbb{R}}_{+};{\mathbb{C}}_p)$ and $H_p^s({\mathbb{R}}_{+};{\mathbb{C}}_p)$ are the natural spaces for a rigorous theory. The real positive proportional spaces remain useful subspaces, especially for applications in which positivity is part of the model, but they are not invariant under the proportional Fourier transform in general.

Scope and Limitations

The approach has four explicit limitations. First, because the theory is conjugate to classical Fourier analysis through ${\mathcal{R}}_p$, the principal analytic theorems are transported rather than independent new Fourier theorems. Second, ${\mathbb{C}}_p$ is a formal topological field copy of $\mathbb{C}$; it is useful for internal notation but does not create additional complex-analytic information. Third, the differential-equation examples are theoretical model problems and do not yet establish an empirical, numerical-complexity, or data-analysis advantage over working directly with logarithmic representatives. Fourth, the present article treats $s\ge 0$ and does not develop proportional tempered distributions or negative-order Sobolev spaces. The value of the framework is therefore a coherent proportional formulation: positive-scale variables, proportional operations, kernels, convolution, and function spaces remain explicit. Stronger future claims require a model in which the proportional variables yield a distinct interpretation, invariant, estimate, or computational benefit that is not merely a relabeling of the representative problem.

The approach suggests a broader program. Proportional arithmetic is one instance of a non-Diophantine arithmetic generated by a bijection. Other bijections can produce different arithmetics, each with its own derivative, integral, function spaces and transform theory. The present work provides a model case in which a core harmonic-analysis package can be constructed while keeping the alternative arithmetic visible at every stage.

11. Conclusions

In this work we developed a proportional Fourier framework arising from proportional arithmetic as a non-Diophantine arithmetic on ${\mathbb{R}}_{+}$. Starting from the operations

$$\oplus ,\ominus ,\odot ,\oslash ,$$

we constructed proportional functions, proportional differentiation, proportional integration, proportional complex numbers and the proportional exponential kernel. These elements allowed us to define the proportional Fourier transform within the proportional setting.

The construction keeps proportional arithmetic visible at every stage. The classical product, ordinary integral and oscillatory exponential are replaced by the proportional product, the proportional integral and the proportional kernel

$$K_p(t,\xi )={\mathrm{Exp}}_p(\ominus i_p\odot \xi \odot t).$$

The representative map is then used to verify the analytic content of the theory. The correspondence theorem shows that the representative of ${\mathcal{F}}_p\left\{f_p\right\}$ is the classical Fourier transform of the representative of $f_p$. This result provides the bridge needed to prove inversion, the Riemann–Lebesgue property, Plancherel’s theorem and the unitary extension of ${\mathcal{F}}_p$ to $L_p^2({\mathbb{R}}_{+};{\mathbb{C}}_p)$.

We also introduced complex-valued proportional Schwartz and Sobolev spaces. The complex formulation is necessary because real proportional inputs may have complex proportional Fourier transforms. The proportional Fourier transform maps ${\mathcal{S}}_p({\mathbb{R}}_{+};{\mathbb{C}}_p)$ bijectively onto itself and characterizes $H_p^s({\mathbb{R}}_{+};{\mathbb{C}}_p)$ regularity through the proportional frequency domain. These results give a functional-analytic framework for studying proportional smoothness, decay and regularity.

The operational rules obtained for ${\mathcal{F}}_p$, including linearity, translation, modulation, scaling, differentiation and convolution, confirm that the proportional Fourier transform preserves the essential structural behavior of the classical Fourier transform while remaining written in the proportional algebra. The weak-derivative and semigroup formulations ensure that the Sobolev and heat-equation statements are not restricted to classically smooth proportional functions. The examples, including proportional Gaussian-type functions, a genuinely complex transform pair, a proportional resolvent equation and a proportional heat equation, illustrate how differential equations formulated in proportional arithmetic can be analyzed systematically.

The relation with Mellin-type analysis and Fourier analysis on multiplicative structures is not ignored. Rather, it is placed in its proper role: the representative map gives a rigorous classical bridge, while the proportional operations preserve the arithmetic structure in which the original objects are formulated. The main outcome is therefore a systematic proportional-arithmetic realization of the core Fourier package on ${\mathbb{R}}_{+}$, including complex scalars, proportional kernels, $L^2$-unitarity, Schwartz invariance, Sobolev regularity and model proportional differential equations.

Future work may proceed in several concrete directions. A first problem is to characterize the dual space ${\mathcal{S}}_p^{\prime }({\mathbb{R}}_{+};{\mathbb{C}}_p)$ through the representative map and to prove that ${\mathcal{F}}_p$ is an automorphism of proportional tempered distributions. This requires specifying the transported topological duality on ${\mathcal{S}}_p$ and proving the continuity of ${\mathcal{F}}_p$ and ${\mathcal{F}}_p^{-1}$ in the corresponding locally convex topology. A second problem is to obtain $L_p^q$-boundedness criteria for proportional Fourier multipliers by transporting classical multiplier conditions and then rewriting them in proportional frequency notation; the classical multiplier theory provides the natural benchmark [16]. A third problem is to formulate proportional pseudo-differential operators of the form ${\mathcal{F}}_p^{-1}\{a_p\odot {\mathcal{F}}_p{f}_p\}$ and identify symbol classes that remain meaningful in proportional variables. It would also be useful to explore other non-Diophantine arithmetics induced by different bijections, since each such arithmetic may generate its own differential, integral and spectral theory.