The Master Integral Transform with Entire Kernels

Abstract

We study an integral transform—here called the Master Integral Transform—in which the kernel is an arbitrary entire function of finite order. When the nonzero Taylor coefficients of the kernel have positive Beurling–Malliavin density, we prove completeness and global injectivity in a Cauchy-weighted Hilbert space, and we furnish explicit Mellin–Fourier inversion formulae with exponentially decaying integrands. Classical Fourier, Laplace, and Mellin transforms appear only as strict special cases. Beyond these, we establish structural properties (multiplier/composition law, dilation covariance, parameter regularity) and present applications not captured by fixed-kernel frameworks, including inverse-kernel identification and hybrid boundary value models, e.g., the Poisson–Airy pair produces a closed-form transformed Green’s function and a solvable variable-coefficient PDE, illustrating capabilities unavailable to fixed-kernel frameworks.

1. Introduction

1.1. Classical Setting and Its Limitations

Integral transforms form the backbone of modern analysis. The Fourier, Laplace, and Mellin transforms, originating in the seminal works of Fourier [1], Laplace [2], and Mellin [3], remain indispensable in harmonic analysis, signal processing, and differential equations; see Titchmarsh [4], Bracewell [5], and Grafakos [6] for surveys [7,8,9,10,11,12].

A unifying feature of these canonical transforms is that each employs a fixed kernel—respectively, $e^{iux}$, $e^{-sx}$, and $x^{s-1}$—tailored to a narrow class of boundary conditions or geometries. Extensions such as fractional Fourier transforms, Hankel transforms, and wavelets relax either the exponent or the geometry, but none permit the analyst to choose an arbitrary analytic kernel while retaining rigorous injectivity and an explicit inversion formula [13].

Two practical consequences follow:

(1)

Inverse-kernel problems—determining the kernel of an integral operator from measured data—cannot be phrased within the classical hierarchy;

(2)

The analyst is often forced to fit non-standard data into the Procrustean bed of Fourier/Laplace kernels, incurring artificial windowing, padding, or ad hoc regularisation.

These limitations motivate the search for a truly adaptive transform theory in which the kernel is variable yet the core analytic structure survives.

1.2. Aim and Guiding Principle

The present paper introduces the Master Integral Transform (MIT), an integral transform whose kernel g is an arbitrary entire function of finite order. A single weighted integral,

$${\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\left(\theta \right)={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{\left|x\right|}^{p}f\left(x\right)e^{-i{\displaystyle \frac{\pi p}{2}}sgnx}g\left(\alpha +\beta e^{i\theta x}\right)dx,$$

recovers Fourier, Laplace, and Mellin transforms for the specific choice $g\left(z\right)=z$ and suitable $(\alpha ,\beta ,p)$.

The guiding principle of the MIT is

$$\mathit{kernel}–\mathit{signalduality}:f\left(x\right)⟺g\left(\alpha +\beta e^{i\theta x}\right),$$

i.e., data may be attached to either argument of the integral. In particular, fixing f and varying g turns the MIT into a natural framework for inverse-kernel identification.

Motivating Example (Hybrid BVP)

Let $f\left(x\right)={\displaystyle \frac{y/\pi }{x^2+y^2}}$ (Poisson kernel) and take the kernel $g\left(z\right)=Ai\left(z\right)$. With $(p,\alpha ,\beta )=(0,0,1)$, we obtain ${\mathcal{G}}_{0,1}^0\left[f\right]\left(\theta \right)={\displaystyle \frac{1}{2\pi }}Ai\left(e^{-\left|\theta \right|y}\right)$, and $\Psi (y,\theta )$ solves a linear PDE with variable coefficients (see Section 9). This is outside the reach of fixed-kernel transforms and exemplifies kernel–signal duality.

Remark 1

(On first appearance). The Master Integral Transform (MIT) is introduced here in its present form. An earlier draft circulated under the working title “Abu-Ghuwaleh Transform (AGT)”; the current paper supersedes that draft and expands the theory (density-driven injectivity, Mellin–Fourier inversion, and applications).

1.3. Main Results

Let $S=\{k\in \mathbb{N}:c_k\ne 0\}$ be the index set of the non-vanishing Taylor coefficients of g. Our analysis hinges on the Beurling–Malliavin (BM) densities

$${\rho }_{*}:=\underset{R\to \infty }{lim\; inf}{\displaystyle \frac{\#(S\cap [0,R\left]\right)}{R}},{\rho }^{*}:=\underset{R>0}{sup}{\displaystyle \frac{\#(S\cap [0,R\left]\right)}{R}}.$$

(R1)

Completeness and frames. If ${\rho }_{*}>0$, then the kernel orbit $\{g(\alpha +\beta e^{i\theta x}):\theta \in \mathbb{R}\}$ is complete in the Cauchy-weighted space $L_{*}^2\left(\mathbb{R}\right)$ (Section 6). If, moreover, ${\rho }^{*}<\infty$, then the orbit forms a frame on any finite window, yielding stability constants.

(R2)

Global injectivity. Under the same lower-density hypothesis, ${\mathcal{G}}_{\alpha ,\beta }^p$ is injective; hence, the kernel orbit carries full information about the signal (Section 6, Theorem 3).

(R3)

Explicit inversion. A Mellin–Fourier contour argument produces cosine and sine inversion formulae whose integrands decay exponentially in the Mellin variable, guaranteeing spectral accuracy in numerical quadrature (Section 7, Theorems 4 and 5).

The BM-density condition is sharp: for $g\left(z\right)=z$, one has ${\rho }_{*}=0$ and the general theorem does not apply, yet classical Fourier injectivity is recovered by the Riemann–Lebesgue theory—illustrating that the MIT strictly extends, rather than duplicates, the classical setting.

1.4. Relationship to Previous Work

The BM theorem for exponential systems goes back to [14]; subsequent refinements are due to Havin–Seip, Nazarov, and others. Our use of BM density differs in two crucial respects:

1.

It is applied to the nonzero Taylor indices of an arbitrary entire kernel, not to its zero set.

2.

The density enters as a quantitative parameter that controls both completeness and numerical condition numbers.

No existing transform theory—fractional, Dunkl, Gabor, wavelet, or generalised Bessel—offers this level of kernel flexibility together with a quantitative completeness criterion and an explicit inverse.

1.4.1. Relation to Other Generalised Transforms

The Neretin G-transform and the Kosugi–Yamazaki mixed Mellin–Fourier transforms both admit analytic kernels, but in each case, either (i) the kernel is fixed in advance or (ii) inversion relies on the Fourier–Plancherel theorem and thus fails outside $L^2$. By contrast, the MIT (i) lets the kernel be arbitrary entire of finite type; (ii) proves injectivity in the Cauchy-weighted space $L_{*}^2$, which strictly contains $L^2$; and (iii) furnishes an explicit Mellin–Fourier contour inverse with exponential decay, unavailable in the cited theories. Dunkl–Cherednik transforms require reflection-group symmetry and again use a fixed kernel, so they fall outside our duality framework.

1.4.2. Bridge to Cauchy/Toeplitz/Shift Theory

Weighted spaces of Cauchy type, Toeplitz kernels, and shift-invariant structures are classical; see Nikolski [15], Garnett [16], Duren [17], Koosis [18], and Katznelson [19]. Our use here is limited to embedding estimates for $L_{*}^2$ and the density/completeness mechanism via Beurling–Malliavin theory (Levin; Havin–Seip; Makarov–Poltoratski).

1.4.3. What Is New Here vs. Classical

• New: Orbit completeness for entire kernels via BM density of nonzero Taylor indices; global injectivity in $L_{*}^2$; Mellin–Fourier inversion with exponentially decaying integrand; inverse-kernel identification framework (Section 5 and Section 6, Applications).
• Classical: Beurling–Malliavin completeness (Levin; Havin–Seip); Fourier/Mellin identities and standard contour arguments (Titchmarsh; EMOT) [20,21,22].

1.4.4. Authorship and Attribution

Theorems 2 and 3 (orbit completeness for entire kernels via BM density of nonzero Taylor indices, and global injectivity in $L_{*}^2$) and Theorems 4 and 5 (Mellin–Fourier inversion with exponentially decaying integrand) are, to our knowledge, new. Lemma 3 is the classical one-sided Beurling–Malliavin completeness theorem (stated here with a citation to Levin ([23], Chap. VI, Thm. 6)); Remark 9 on frame behaviour follows Havin–Seip [24]. The non-injectivity mechanism in Proposition 4 uses the Toeplitz-kernel gap theory of Makarov–Poltoratski ([25], Cor. 2). Standard Mellin/Fourier identities are used with [26,27,28]. Dirichlet-series bounds in Lemma 6 follow standard arguments as in ([29], Chap. 12). Where we reproduce classical statements, we cite the original sources at the point of use.

1.5. Structure of This Paper

Section 2 settles notation and recalls the necessary Beurling–Malliavin background. Section 3 defines the MIT and its even/odd branches. Classical transforms are recovered in Section 5. Completeness, frames, and injectivity occupy Section 6. Section 7 presents the Mellin–Fourier inversion engine. Section 9 contains the application suite, after which Section 10 lists major open questions. An extensive MAG catalogue and three auxiliary inversion tool kits are deferred to the appendices.

2. Preliminaries and Basic Assumptions

Throughout this paper, we fix a complex weight exponent p with

$$-1<\Re p<1,$$

and two kernel parameters $\alpha ,\beta \in \mathbb{C}$ with $\beta \ne 0$. All constants are independent of the transform’s spectral parameter $\theta \in \mathbb{R}$ unless stated otherwise.

2.1. Cauchy-Weighted Hilbert Space

Definition 1

(Cauchy weight). Define

$$L_{*}^2\left(\mathbb{R}\right):=L^2\left(\mathbb{R},{(1+x^2)}^{-1}dx\right),{⟨F,G⟩}_{*}:={\int }_{\mathbb{R}}F\left(x\right)\overline{G\left(x\right)}{\displaystyle \frac{dx}{1+x^2}}.$$

Remark 2

(Dense, continuous embedding of $L^2$). For every $A>0$ we have ${(1+A^2)}^{-1}{\parallel h\parallel }_{L^2(-A,A)}^2\le {\parallel h\parallel }_{L_{*}^2}^2\le {\parallel h\parallel }_{L^2(-A,A)}^2.$ Hence, the identity map $I:L^2\left(\mathbb{R}\right)\to L_{*}^2\left(\mathbb{R}\right)$ is continuous and has dense range. Throughout the sequel, inner products $⟨·,·⟩$ may therefore be taken in either space without ambiguity.

2.2. p-Admissible Signals

Assumption 1

(p-admissible functions). A measurable $f:\mathbb{R}\to \mathbb{C}$ is p-admissible if

$${\left|x\right|}^{\Re p}f\left(x\right)\in L^1\left(\mathbb{R}\right),{\left|x\right|}^{p}f\left(x\right)e^{-i{\displaystyle \frac{\pi p}{2}}sgnx}\in L_{*}^2\left(\mathbb{R}\right),$$

(1)

and there exists $m>1+\Re p$ such that $f\left(x\right)=O\left({\left|x\right|}^{-m}\right)$ as $\left|x\right|\to \infty$.

Remark 3

(Integrability hierarchy). Because $-1<\Re p<1$, the algebraic weight in (1) is weaker than the Cauchy weight; consequently, every p-admissible f lies in $L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right)\subset L_{*}^2\left(\mathbb{R}\right)$.

2.3. Entire Kernels and BM Density

Assumption 2

(Entire kernel with BM bounds). Let $g:\mathbb{C}\to \mathbb{C}$ be an entire function of finite order $\rho <\infty$. Expand about α:

$$g(\alpha +z)=\sum _{k=0}^{\infty }{c}_k{z}^k,S:=\{k\in \mathbb{N}:c_k\ne 0\}.$$

Assume the two-sided Beurling–Malliavin inequalities

$$0<{\rho }_{*}:=\underset{R\to \infty }{lim\; inf}{\displaystyle \frac{\#\{k\in S:k\le R\}}{R}}\le {\rho }^{*}:=\underset{R>0}{sup}{\displaystyle \frac{\#\{k\in S:k\le R\}}{R}}<\infty .$$

(2)

Finally, choose radii $0<r_{-}<\left|\beta \right|<r_{+}$ so that $g\in H^{\infty }\left(\{r_{-}<|z-\alpha |<r_{+}\}\right).$

Assumption 3

(Normalisation of the first two Taylor coefficients). Write $c_k=g^{\left(k\right)}\left(\alpha \right)/k!$. We assume $c_0=0$ and $c_1\ne 0$. If $c_0\ne 0$, then one may replace $g\left(z\right)$ by $g\left(z\right)-g\left(\alpha \right)$, which shifts the transform by a constant multiple of $\int \Phi$ but leaves all proofs intact.

Lemma 1

(Exponential coefficient bound). For every such $r_{+}$ there exists $C=C(g,r_{+})$ with $|c_k|\le C{r}_{+}^{-k}$$(k\ge 0)$.

Proof. 

Apply Cauchy’s estimate to the circle $|z-\alpha |=r_{+}$ and the $H^{\infty }$ boundedness of g on that circle. □

Remark 4

(Choice of $r_{\pm }$). Because $\beta \ne 0$, one may enlarge $r_{+}$ slightly so that $\left|\beta \right|<r_{+}$. Uniform boundedness of g on closed sub-annuli is used in all forthcoming $L^2$ estimates.

2.4. Notation Summary

We collect recurring symbols in Table 1 for quick reference.

Table 1. Notation used throughout the paper.

Symbol	Description
p	Algebraic weight exponent ($-1<\Re p<1$)
$\alpha ,\beta$	Kernel shift and scale ($\beta \ne 0$)
g	Entire kernel satisfying Assumption 2
${\rho }_{*},{\rho }^{*}$	Lower/upper BM densities, Equation (2)
$L_{*}^2\left(\mathbb{R}\right)$	Cauchy-weighted Hilbert space, Definition 1
$\Phi$	Weighted signal ${\left|x\right|}^{p}f\left(x\right)e^{-i{\displaystyle \frac{\pi p}{2}}sgnx}$
$\widehat{\Phi }$	Fourier transform of $\Phi$ on $\mathbb{R}$

Section 3 introduces the Master Integral Transform itself and establishes its fundamental properties.

3. The Master Integral Transform

This section introduces the one-parameter transform that stands at the centre of the present work. We first give a precise definition, then record elementary properties that will be needed throughout this paper.

3.1. Definition and Sign Conventions

Definition 2

(Master Integral Transform (MIT)). Let $-1<\Re p<1$ and let f be p-admissible (Assumption 1). Let g be entire of finite order, with nonzero Taylor index set S having positive lower Beurling–Malliavin density, and assume g is bounded on an annulus $r_{-}<|z-\alpha |<r_{+}$ with $\left|\beta \right|<r_{+}$. For $\theta \in \mathbb{R}$, define

$${\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\left(\theta \right):={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}{\left|x\right|}^{p}f\left(x\right)e^{-i{\displaystyle \frac{\pi p}{2}}sgnx}g\left(\alpha +\beta e^{i\theta x}\right)dx.$$

(3)

This maps the p-admissible class ${\mathcal{D}}_p\subset L^1\cap L^2$ into functions that admit the strip analyticity described in Proposition 1; absolute convergence holds by Lemma 2.

Parameter Dictionary and Domain (At a Glance)

Here $\alpha ,\beta \in \mathbb{C}$ with $\beta \ne 0$ are the kernel shift/scale; $p\in \mathbb{C}$ with $-1<\Re p<1$ is the algebraic weight exponent; and $\theta \in \mathbb{R}$ is the spectral parameter. The kernel g is entire of finite order and satisfies the BM-density hypothesis in Assumption 2. The map (3) acts on the p-admissible class ${\mathcal{D}}_p$ of Assumption 1 and lands in $L_{*}^2\left(\mathbb{R}\right)$ with absolute convergence by Lemma 2. We use the principal branch for $z^p$ (Remark 5), and $sgn(·)$ denotes the usual sign function.

Remark 5

(Branch of the algebraic phase). Throughout this paper, we fix the principal branch of $z^p=e^{plogz}$ with $logz=ln\left|z\right|+iargz$ and $-\pi <argz<\pi$. Consequently, the factor $e^{-i{\displaystyle \frac{\pi p}{2}}sgnx}$ in (3) is single-valued for every $x\ne 0$.

Lemma 2

(Absolute convergence). If f is p-admissible, then the integral in (3) converges absolutely for every $\theta \in \mathbb{R}$.

Proof. 

Split the x-axis at $\left|x\right|=1$. On $\left|x\right|\le 1$, we have $\left|g\right|\le {\parallel g\parallel }_{H^{\infty }}$ by Assumption 2, and the integrand is in $L^1\left((-1,1)\right)$ by (1). For $\left|x\right|\ge 1$, the decay $f\left(x\right)=O\left(\right|x{|}^{-m})$ with $m>1+\Re p$ implies ${\left|x\right|}^{\Re p}\left|f\left(x\right)\right|={O\left(\right|x|}^{-1-\epsilon })$ for some $\epsilon >0$, so the tail integral converges absolutely. □

Remark 6

(On the sign factor). The phase $e^{-i{\displaystyle \frac{\pi p}{2}}sgnx}$ is the standard choice consistent with the principal branch of $z^p$; it ensures the algebraic weight is single-valued on $\mathbb{R}\setminus \left\{0\right\}$ and is natural for hybrid even/odd decompositions.

3.2. Analytic Continuation in $\theta$

Proposition 1

(Strip of analyticity). Fix $x\ne 0$. For $|\Im \theta |<\left(ln(r_{+}/|\beta \left|\right)\right)/\left|x\right|$, the map $\theta \mapsto g(\alpha +\beta e^{i\theta x})$ is analytic; hence, $\theta \mapsto {\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\left(\theta \right)$ extends holomorphically to any vertical strip contained in the intersection of these pointwise strips. No uniform strip is guaranteed unless additional decay in x is assumed.

Proof. 

Immediate from the definition and the maximum modulus principle on the annulus, $\{r_{-}<|z-\alpha |<r_{+}\}$. □

3.3. Even/Odd Decomposition

Write $f=f_{\mathrm{e}}+f_{\mathrm{o}}$ with

$$f_{\mathrm{e}}\left(x\right)=\frac{1}{2}\left(f\left(x\right)+f(-x)\right),f_{\mathrm{o}}\left(x\right)=\frac{1}{2}\left(f\left(x\right)-f(-x)\right).$$

Split the integral at $x=0$ and use ${\left|x\right|}^p=x^p$ for $x>0$ to obtain

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{c}}\left[f\right]\left(\theta \right)& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }{x}^p{f}_{\mathrm{e}}\left(x\right)\left[e^{-i{\displaystyle \frac{\pi p}{2}}}g(\alpha +\beta e^{i\theta x})+e^{i{\displaystyle \frac{\pi p}{2}}}g(\alpha +\beta e^{-i\theta x})\right]dx,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{s}}\left[f\right]\left(\theta \right)& ={\displaystyle \frac{1}{i\pi }}{\int }_0^{\infty }{x}^p{f}_{\mathrm{o}}\left(x\right)\left[e^{-i{\displaystyle \frac{\pi p}{2}}}g(\alpha +\beta e^{i\theta x})-e^{i{\displaystyle \frac{\pi p}{2}}}g(\alpha +\beta e^{-i\theta x})\right]dx,\hfill \end{array}$$

(5)

the cosine and sine branches of the MIT. They enjoy the reconstruction identity

$${\mathcal{G}}_{\alpha ,\beta }^p\left[f\right](\pm \theta )=\frac{1}{2}{\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{c}}\left[f_{\mathrm{e}}\right]\left(\theta \right)\pm \frac{i}{2}{\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{s}}\left[f_{\mathrm{o}}\right]\left(\theta \right),\theta >0.$$

(6)

Remark 7

(Symmetry). For even signals ($f=f_{\mathrm{e}}$), the transform is an even function of θ, whereas for odd signals ($f=f_{\mathrm{o}}$) it is odd, matching the classical Fourier cosine/sine dichotomy at $(\alpha ,\beta ,p)=(0,1,0)$

3.4. Kernel–Signal Duality

Observe that

$$f\left(x\right)⟷g\left(\alpha +\beta e^{i\theta x}\right)$$

enter (3) on an exactly symmetrical footing. Hence, the MIT can be viewed either as a transform

$$f⟼{\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\mathrm{with}\mathrm{kernel}\mathrm{fixed},$$

or, in the inverse-kernel direction,

$$g⟼{\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\mathrm{with}\mathrm{the}\mathrm{signal}f\mathrm{fixed}.$$

Section 6 shows that when the kernel satisfies the Beurling–Malliavin condition (2), the mapping $f\mapsto {\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]$ is injective; its left-inverse is constructed explicitly in Section 7.

Section 5 verifies that the Fourier, Laplace, and Mellin transforms emerge as strict special cases of (3)–(5).

4. Structural Properties of the Master Integral Transform

Definition 3

(Dirichlet factor of the kernel). Let $g(\alpha +z)={\sum }_{k\ge 0}{c}_k{z}^k$ with $c_1\ne 0$ and $S=\{k\ge 1:c_k\ne 0\}$. Define

$$D\left(s\right):=\sum _{k\in S}{c}_k{\beta }^k{k}^{-s},s\in \mathbb{C}.$$

Theorem 1

(Multiplier/composition law). For $f\in {\mathcal{D}}_p$ and $\Re s>0$,

$${\mathcal{M}}_{\theta }\left\{{\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{c}}\left[f\right]\right\}\left(s\right)={\displaystyle \frac{\Gamma \left(s\right)cos(\pi s/2)}{\pi }}D\left(s\right)\mathcal{M}\left\{{\Phi }_{\mathrm{e}}\right\}(1-s),$$

$${\mathcal{M}}_{\theta }\left\{{\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{s}}\left[f\right]\right\}\left(s\right)={\displaystyle \frac{\Gamma \left(s\right)sin(\pi s/2)}{i\pi }}D\left(s\right)\mathcal{M}\left\{{\Phi }_{\mathrm{o}}\right\}(1-s),$$

with ${\Phi }_{\mathrm{e}/o}$ as in (4) and (5). Hence, the composition of MIT operators corresponds to multiplication by $D\left(s\right)$ in the θ–Mellin domain.

Proposition 2

(Dilation covariance). For $a>0$, let $f_a\left(x\right)=f\left(ax\right)$. Then

$${\mathcal{G}}_{\alpha ,\beta }^p\left[f_a\right]\left(\theta \right)=a^{-1-p}{\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\left(\frac{\theta }{a}\right).$$

In particular, injectivity and frame properties are preserved under dilations.

Proposition 3

(Parameter regularity). Assume g is entire of finite order and bounded on an annulus $r_{-}<|z-\alpha |<r_{+}$ with $\left|\beta \right|<r_{+}$. Then, the following hold:

(a) 

$(\alpha ,\beta )\mapsto {\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\left(\theta \right)$ is holomorphic on $\{(\alpha ,\beta ):r_{-}<|z-\alpha |<r_{+},|\beta |<r_{+}\}$ for each fixed θ;

(b) 

For every compact vertical strip in θ, ${\partial }_{\theta }^m{\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\left(\theta \right)$ exists for all $m\ge 0$ and satisfies Gevrey-type bounds depending only on g, p, and the strip.

5. Classical Transforms as Specialisations

The classical transforms recovered by the MIT are summarized in Table 2.

Table 2. Recovering classical transforms (analytic continuation in $\theta$ is used for Laplace and Mellin.

Transform	$\mathit{\alpha }$	$\mathit{\beta }$	p	$\mathit{g}\left(\mathit{z}\right)$	MIT Formula
Two-sided Fourier	0	1	0	z	$\mathcal{F}f\left(\theta \right)=2\pi {\mathcal{G}}_{0,1}^0\left[f\right]\left(\theta \right)$
Bilateral Laplace	0	1	0	z	$\mathcal{L}f\left(s\right)=2\pi {\mathcal{G}}_{0,1}^0\left[f\right]\left(is\right)$
Mellin ($x>0$, $\theta =0$)	0	1	$s-1$	z	${\displaystyle \mathcal{M}f\left(s\right)=2\pi e^{i\frac{\pi (s-1)}{2}}{\mathcal{G}}_{0,1}^{s-1}\left[f\right]\left(0\right)}$
Fourier cosine (half)	0	1	0	z	cos part of (4)
Fourier sine (half)	0	1	0	z	sine part of (5)

Proof. 

(Verification (Fourier row)) Insert $(\alpha ,\beta ,p,g)=(0,1,0,z)$ into (3) to obtain ${\mathcal{G}}_{0,1}^0\left[f\right]\left(\theta \right)={\displaystyle \frac{1}{2\pi }}\widehat{f}\left(\theta \right).$ Multiplying by $2\pi$ yields the standard Fourier transform. The other rows follow analogously. □

Remark 8

(Injectivity for the classical cases). The kernel $g\left(z\right)=z$, used to recover the Fourier and Laplace transforms, has a set of nonzero Taylor coefficients $S=\left\{1\right\}$. This set has a Beurling–Malliavin density of zero. Consequently, our main injectivity result (Theorem 3), which requires positive density, does not apply to these special cases. However, their injectivity is already well-established by the classical theories of the Fourier and Laplace transforms themselves.

For the bilateral Laplace line, we require $\Re s>0$ to guarantee absolute convergence of the defining integral.

Recovering the Mellin Transform

Choose $\alpha =0$, $\beta =1$, put $p=s-1$ with $\Re \left(s\right)>0$, and take the kernel $g\left(z\right)=z$. Formula (3) then becomes

$${\mathcal{G}}_{0,1}^{s-1}\left[f\right]\left(\theta \right)={\displaystyle \frac{e^{-i\frac{\pi (s-1)}{2}}}{2\pi }}{\int }_0^{\infty }{x}^{s-1}f\left(x\right)e^{i\theta x}dx,(\theta \in \mathbb{R},x>0),$$

where we have restricted to $x>0$ because the classical Mellin transform is one-sided. Setting $\theta =0$ gives

$${\mathcal{G}}_{0,1}^{s-1}\left[f\right]\left(0\right)={\displaystyle \frac{e^{-i\frac{\pi (s-1)}{2}}}{2\pi }}{\int }_0^{\infty }{x}^{s-1}f\left(x\right)dx={\displaystyle \frac{e^{-i\frac{\pi (s-1)}{2}}}{2\pi }}\mathcal{M}f\left(s\right),$$

so that $\mathcal{M}f\left(s\right)=2\pi e^{i{\displaystyle \frac{\pi (s-1)}{2}}}{\mathcal{G}}_{0,1}^{s-1}\left[f\right]\left(0\right).$ Thus, the Master Integral Transform reproduces the Mellin transform exactly—up to the harmless global phase $e^{i\pi (s-1)/2}$ that varies among Mellin conventions.

The cosine and sine transforms in the last two rows follow by inserting $f=f_{\mathrm{e}}$ or $f=f_{\mathrm{o}}$ into (4) and (5) and taking $(\alpha ,\beta ,p)=(0,1,0)$.

6. Completeness, Frames, and Injectivity

The Master Integral Transform is meaningful for any entire kernel, but invertibility requires a quantitative condition on the set of nonzero Taylor coefficients. In this section, we prove this condition, state precise completeness and frame results, and derive global injectivity.

Throughout, we retain the standing data of Section 2 and impose one additional—and essentially sharp—normalisation on the kernel.

6.1. BM Completeness for Sparse Exponentials

Lemma 3

(Beurling–Malliavin completeness). Let $S\subset \mathbb{N}$ satisfy ${\rho }_{*}>0$. If $\theta \ne 0$ and $A<\pi {\rho }_{*}/\left|\theta \right|$, then the exponential system ${\left\{e^{ik\theta x}\right\}}_{k\in S}$ is complete in $L^2(-A,A)$; that is, no nonzero $F\in L^2(-A,A)$ can be orthogonal to every element of the system.

Proof. 

This is the one-sided Beurling–Malliavin completeness theorem, see Levin ([23], Chap. VI, Thm 6). (Doubling the set to ${\{\pm k\theta \}}_{k\in S}$ reduces to the classical two-sided statement, but the one-sided form suffices.) □

6.2. A Fourier-Coefficient Annihilation Lemma

Lemma 4

(Key linear system). Fix $\theta \ne 0$ and let $F\in L_{*}^2\left(\mathbb{R}\right)$ satisfy

$${⟨F,g\left(\alpha +\beta e^{i\theta ·}\right)⟩}_{*}=0.$$

Denote the (unweighted) Fourier coefficients $a_n:={⟨F,e^{in\theta ·}⟩}_{L^2}(n\in \mathbb{Z}).$ Then for every $k\in S=\{m:c_m\ne 0\}$, one has

$$c_1\beta a_k+\sum _{m=2}^{\infty }{c}_m{\beta }^m{a}_{mk}=0.$$

(7)

Moreover, choosing $r_{+}$ large enough so that

$$r_{+}>{\displaystyle \frac{2C}{|c_1|}},q:={\displaystyle \frac{\left|\beta \right|}{r_{+}}}<{\displaystyle \frac{|c_1\beta |}{2C}}\le {\displaystyle \frac{1}{2}},$$

(8)

forces $a_k=0$ for all $k\in S$.

Proof. 

Step 1: Kernel expansion and absolute convergence. With the Taylor series $g(\alpha +\beta e^{i\theta x})={\sum }_{m\ge 0}{c}_m{\beta }^m{e}^{im\theta x}$, we get

$${⟨F,g(\alpha +\beta e^{i\theta ·})⟩}_{*}=\sum _{m=0}^{\infty }{c}_m{\beta }^m{⟨F,e^{im\theta ·}⟩}_{*}.$$

Lemma 1 gives $|c_m|\le C{r}_{+}^{-m}$, so the series is absolutely convergent when $q=\left|\beta \right|/r_{+}<1$, and Tonelli’s theorem allows us to interchange sum and inner product.

Step 2: Switch to $L^2$ coefficients. Because $L^2\hookrightarrow L_{*}^2$ densely, the inner products coincide with $a_m$, yielding (7) for every $k\in S$.

Step 3: Contraction argument with explicit constants. With the radius choice (8), $q\le {\displaystyle \frac{1}{2}}$. Taking absolute values in (7) and using $|c_m|\le C{r}_{+}^{-m}=C{q}^m{\left|\beta \right|}^{-m}$,

$$|c_1\beta ||a_k|\le C\sum _{m\ge 2}{q}^m|a_{mk}|\le C{\displaystyle \frac{q^2}{1-q}}\underset{m\ge 2}{sup}|a_{mk}|.$$

Since $q\le {\displaystyle \frac{1}{2}}$, the factor $q^2/(1-q)\le \frac{1}{3}<1$. Iterating along the chain $k\mapsto 2k\mapsto 4k\mapsto \dots$ drives $|a_k|\to 0$; hence, $a_k=0$ for every $k\in S$. □

6.3. Completeness of the Kernel Orbit

Theorem 2

(Orbit completeness). Under Assumptions 2 and 3, the set

$${\Theta }_g:=\left\{x\mapsto g\left(\alpha +\beta e^{i\theta x}\right):\theta \in \mathbb{R}\right\}\subset L_{*}^2\left(\mathbb{R}\right)$$

is complete: the only function orthogonal to every element of ${\Theta }_g$ is 0.

Proof. 

Assume $F\in L_{*}^2\left(\mathbb{R}\right)$ is orthogonal to ${\Theta }_g$. Choose a sequence ${\theta }_n\to 0$ with $A_n:=\pi {\rho }_{*}/\left|{\theta }_n\right|-\epsilon \to \infty$ ($\epsilon >0$). For each n, Lemma 4 yields $a_k^{\left(n\right)}=0$ for all $k\in S$, where $a_k^{\left(n\right)}:={⟨F,e^{ik{\theta }_n·}⟩}_{*}$. By Lemma 3, the system ${\left\{e^{ik{\theta }_{n}x}\right\}}_{k\in S}$ is complete in $L^2(-A_n,A_n)$, hence $F=0$ a.e. on $(-A_n,A_n)$. Letting $n\to \infty$ gives $F=0$ on $\mathbb{R}$. □

Remark 9

(Frame bounds on every finite window). Because ${\rho }^{*}<\infty$, the same BM theory [24] implies that for each $A>0$, the orbit restricted to $(-A,A)$ is a frame in $L^2(-A,A)$. Consequently, the MIT is not merely injective but stably invertible on finite intervals.

6.4. Global Injectivity of the MIT

Theorem 3

(Injectivity). Let f be p-admissible and assume the kernel g satisfies Assumptions 2 and 3. If ${\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\left(\theta \right)\equiv 0\mathrm{for}\mathrm{every}\theta \in \mathbb{R},$ then $f=0$ almost everywhere.

Proof. 

Define $\Phi \left(x\right)={\left|x\right|}^{p}f\left(x\right)e^{-i{\displaystyle \frac{\pi p}{2}}sgnx}\in L_{*}^2\left(\mathbb{R}\right)$. The hypothesis says precisely that $\Phi$ is orthogonal to the orbit ${\Theta }_g$. By Theorem 2, this forces $\Phi =0$, hence $f=0$. □

6.5. A Glance at Optimality

If $g\left(z\right)=z$, then $S=\left\{1\right\}$ and ${\rho }_{*}=0$; Theorem 3 no longer applies, yet Fourier and Laplace transforms are classically injective. Thus, the BM condition is sufficient but not necessary. Sharp thresholds on ${\rho }_{*}$ remain an open problem (see Open Problem (B) in Section 10).

Example 1

(Injective transform with ${\rho }_{*}=0$). Take $g\left(z\right)=z$ so $S=\left\{1\right\}$ and ${\rho }_{*}=0$. With $(\alpha ,\beta ,p)=(0,1,0)$, the MIT reduces to the two-sided Fourier transform, which is injective on $L^1\cap L^2$ by the Riemann–Lebesgue lemma. Thus, ${\rho }_{*}>0$ is not necessary.

Proposition 4

(Non-injectivity when ${\rho }_{*}=0$). Let $g\left(z\right)={\sum }_{k\ge 1}{z}^{k!}$ (Taylor support $\{1!,2!,3!,\dots \}$). Then ${\rho }_{*}=0$. With $(\alpha ,\beta ,p)=(0,1,0)$ there exists a nonzero $f\in C_c^{\infty }\left(\mathbb{R}\right)$ such that ${\mathcal{G}}_{0,1}^0\left[f\right]\equiv 0$.

Proof. 

Because g is lacunary, the set ${\left\{g\left(e^{i\theta x}\right)\right\}}_{\theta \in \mathbb{R}}$ fails the gap condition of Makarov–Poltoratski ([25], Cor. 2). Hence, the orbit is not complete and injectivity fails. □

This shows that a positive lower BM density is not merely sufficient but, in a quantitative sense, “almost necessary”. The next section constructs an explicit left-inverse of the MIT, confirming that injectivity is accompanied by a practical inversion formula with spectral decay.

7. Inversion Theory

The preceding section showed that the Master Integral Transform is injective under a Beurling–Malliavin condition on the kernel. We now construct an explicit left-inverse, obtaining retrievable formulae for the even and odd parts of the signal. All steps are carried out in the Cauchy-weighted Hilbert space $L_{*}^2\left(\mathbb{R}\right)$ introduced in Section 2.

Throughout this section, the kernel g satisfies Assumptions 2 and 3; the signal f is p-admissible (Assumption 1); and the shorthand

$$\Phi \left(x\right):={\left|x\right|}^{p}f\left(x\right)e^{-i{\displaystyle \frac{\pi p}{2}}sgnx}$$

is retained.

7.1. Mellin and Auxiliary Transforms

Definition 4

(Bilateral Mellin transform). For a measurable $\Psi :\mathbb{R}\setminus \left\{0\right\}\to \mathbb{C}$, define

$$\mathcal{M}\{\Psi \}\left(u\right):={\int }_{-\infty }^{\infty }{\left|x\right|}^{u-1}\Psi \left(x\right)dx,u\in \mathbb{C},$$

whenever the integral converges absolutely.

Definition 5

(One-sided Mellin in $\theta$). For a function $H:(0,\infty )\to \mathbb{C}$, we set

$${\mathcal{M}}_{\theta }\left\{H\right\}\left(s\right):={\int }_0^{\infty }{\theta }^{s-1}H\left(\theta \right)d\theta ,s\in \mathbb{C}.$$

Lemma 5

(Fubini admissibility). If f is p-admissible with decay exponent $m>1+\Re p$, then every double integral in this section is absolutely convergent; sums and integrals may be interchanged without further comment.

Proof. 

For $\left|x\right|\ge 1$, we have $\left|f\left(x\right)\right|\le {C\left|x\right|}^{-m}$ with $m>1+\Re p$, hence ${\left|x\right|}^{\Re p}\left|f\left(x\right)\right|\le {C\left|x\right|}^{-1-\epsilon }$ for some $\epsilon >0$. Together with $|c_k|\le C{r}_{+}^{-k}$, this yields

$${\int \int |x|}^{\Re p-1-\epsilon }|c_k{||\beta |}^{k}dxdk<\infty .$$

Therefore, every double sum/integral in Section 6 is absolutely convergent and Fubini–Tonelli applies. □

7.2. Dirichlet Series of the Kernel

Definition 6.

Set

$$D\left(s\right):=\sum _{k\in S}{c}_k{\beta }^k{k}^{-s},s\in \mathbb{C}.$$

Lemma 6

(Half-plane of non-vanishing). Assume $c_1\ne 0$ (Assumption 3). There exists ${\sigma }_0\in \mathbb{R}$ such that $D\left(s\right)\ne 0$ and ${\left|D\left(s\right)\right|}^{-1}=O\left({(1+|s\left|\right)}^M\right)$ for all s with $\Re s>{\sigma }_0$ and some $M=M\left(g\right)$.

Proof. 

Absolute convergence for $\Re s>\rho +{log}_{r_{+}}(1/|\beta \left|\right)$ follows from Lemma 1. Since $c_1\ne 0$, the first term dominates when $\Re s$ is large, so $D\left(s\right)$ never vanishes in that half-plane. Polynomial bounds are standard for Dirichlet series with exponentially decaying coefficients. □

7.3. Mellin Transform of the Cosine and Sine Branches

Recall the branches (4) and (5). For the even part, define ${\Phi }_{\mathrm{e}}\left(x\right):=f_{\mathrm{e}}\left(x\right){\left|x\right|}^p{e}^{-i{\displaystyle \frac{\pi p}{2}}sgnx}$ and likewise ${\Phi }_{\mathrm{o}}$.

Lemma 7

(Factorisation identities). For $\Re s>0$ , one has

$$\begin{array}{cc}\hfill {\mathcal{M}}_{\theta }\left\{{\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{c}}\left[f\right]\right\}\left(s\right)& ={\displaystyle \frac{\Gamma \left(s\right)cos(\pi s/2)}{\pi }}D\left(s\right)\mathcal{M}\left\{{\Phi }_{\mathrm{e}}\right\}(1-s),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\mathcal{M}}_{\theta }\left\{{\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{s}}\left[f\right]\right\}\left(s\right)& ={\displaystyle \frac{\Gamma \left(s\right)sin(\pi s/2)}{i\pi }}D\left(s\right)\mathcal{M}\left\{{\Phi }_{\mathrm{o}}\right\}(1-s).\hfill \end{array}$$

(10)

Proof. 

Insert the Taylor series of g, interchange summation and integration (using Lemma 5), and evaluate the $\theta$-integral ${\int }_0^{\infty }{\theta }^{s-1}{e}^{\pm i\theta x}d\theta =\Gamma \left(s\right)e^{\pm i\pi s/2}{\left|x\right|}^{-s}$. □

7.4. Inversion of the Even Branch

Let $u:=1-s$. From (9), we solve for the Mellin transform of ${\Phi }_{\mathrm{e}}$:

$$\mathcal{M}\left\{{\Phi }_{\mathrm{e}}\right\}\left(u\right)={\displaystyle \frac{\pi {\mathcal{M}}_{\theta }\left\{{\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{c}}\left[f\right]\right\}(1-u)}{\Gamma (1-u)sin(\pi u/2)D(1-u)}}.$$

Choose c with

$$0<c<min\left\{1,m-1-\Re p,1-{\sigma }_0\right\},$$

so that all denominators are analytic and $\Gamma (1-u)sin(\pi u/2)$ has no zeros on $\Re u=c$. The bilateral Mellin inversion formula gives the following:

Choice of the Vertical Contour

The Mellin transform $\mathcal{M}\left\{{\Phi }_{\mathrm{e}/o}\right\}\left(u\right)$ is analytic in $0<\Re u<m-1-\Re p$ because ${\left|x\right|}^{\Re u-1}{\Phi }_{\mathrm{e}/o}\left(x\right)\in L^1\left(\mathbb{R}\right)$ for such u. Lemma 6 shows $D(1-u)$ is nonzero for $\Re u<c_0:=1-{\sigma }_0$. Hence, selecting

$$0<c<min\left\{m-1-\Re p,c_0\right\}$$

places the contour $\Re u=c$ inside a pole-free strip; no additional residues arise.

Theorem 4

(Cosine-branch inversion). For every $x\ne 0$

$$f_{\mathrm{e}}\left(x\right)={\displaystyle \frac{e^{i\frac{\pi p}{2}sgnx}}{{\left|x\right|}^p}}{\displaystyle \frac{1}{2\pi i}}{\int }_{c-i\infty }^{c+i\infty }{\left|x\right|}^{-u}{\displaystyle \frac{\pi {\mathcal{M}}_{\theta }\left\{{\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{c}}\left[f\right]\right\}(1-u)}{\Gamma (1-u)sin(\pi u/2)D(1-u)}}du.$$

Remark 10

(Decay of the integrand). Stirling’s formula yields ${|\Gamma (1-u)|}^{-1}=O\left(e^{\pi \left|\tau \right|/2}{\left|\tau \right|}^{\sigma -1/2}\right)$ as $\tau =\Im u\to \infty$, while ${\left|D(1-u)\right|}^{-1}={O\left(\right|\tau |}^M)$ by Lemma 6. Hence, the integrand decays like $e^{-\pi \left|\tau \right|/2}$, giving spectral convergence of simple trapezoidal quadrature rules.

7.5. Inversion of the Odd Branch

Replacing $cos(\pi s/2)$ by $sin(\pi s/2)$ produces

$$\mathcal{M}\left\{{\Phi }_{\mathrm{o}}\right\}\left(u\right)={\displaystyle \frac{i\pi {\mathcal{M}}_{\theta }\left\{{\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{s}}\left[f\right]\right\}(1-u)}{\Gamma (1-u)cos(\pi u/2)D(1-u)}},$$

and exactly the same contour gives the following:

Theorem 5

(Sine-branch inversion). With the same c as in Theorem 4,

$$f_{\mathrm{o}}\left(x\right)={\displaystyle \frac{e^{i\frac{\pi p}{2}sgnx}}{{\left|x\right|}^p}}{\displaystyle \frac{1}{2\pi i}}{\int }_{c-i\infty }^{c+i\infty }{\left|x\right|}^{-u}{\displaystyle \frac{i\pi {\mathcal{M}}_{\theta }\left\{{\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{s}}\left[f\right]\right\}(1-u)}{\Gamma (1-u)cos(\pi u/2)D(1-u)}}du.$$

7.6. Pointwise Reconstruction of f

Corollary 1

(Full inversion). For every $x\ne 0$

$$f\left(x\right)=f_{\mathrm{e}}\left(x\right)+f_{\mathrm{o}}\left(x\right),$$

where $f_{\mathrm{e}}$ and $f_{\mathrm{o}}$ are given by Theorems 4 and 5.

Proof. 

Since $f=f_{\mathrm{e}}+f_{\mathrm{o}}$ and its inversion is linear, add the two contour integrals. □

Remark 11

(Consistency with Section 6). The corollary provides an explicit left-inverse of the MIT; composing it with the forward transform yields the identity on the class of p-admissible functions, exactly matching the injectivity statement of Theorem 3.

Section 9 applies these inversion formulae to a variety of analytic kernels, showcasing problems that are beyond reach of the classical transform triad.

8. General Rational-Weight MAG Transform: Residue and Real-Line Kernel Derivations

8.1. Overview

The next pages present two complementary derivations of a single closed-form formula for the Master Integral Transform when the weight is an arbitrary proper rational function $f_{P,Q}\left(x\right)=P\left(x\right)/Q\left(x\right)$ with $degQ=degP+1$. Section 8.3 follows the classical contour–residue route that arrives at the same general expression (11), from which every entry of the published MAG catalogue is recovered as a special choice of poles and multiplicities.

8.2. Standing Data and Notation

• Analytic kernel g obeys Assumption 2 (entire of finite order, BM density, bounded on a slightly larger annulus).
• Algebraic weight exponent p satisfies $-1<\Re p<1$ to guarantee ${\left|x\right|}^{p}f\left(x\right)\in L^1\cap L_{*}^2$.
• Proper rational weight

  $$f_{P,Q}\left(x\right)={\displaystyle \frac{P\left(x\right)}{Q\left(x\right)}},degQ=degP+1,$$

  with coprime $P,Q\in \mathbb{C}\left[x\right]$ and

  $$Q\left(x\right)=c_0\prod _{k=1}^N{(x-a_k)}^{m_k},a_k\in \mathbb{C}\setminus \mathbb{R},m_k\ge 1.$$
• Residues $r_{k,\ell }:=P^{(\ell -1)}\left(a_k\right)/[(\ell -1)!Q^{\prime }\left(a_k\right)]$ from the partial-fraction expansion

  $$f_{P,Q}\left(x\right)=\sum _{k=1}^N\sum _{\ell =1}^{m_k}{\displaystyle \frac{r_{k,\ell }}{{(x-a_k)}^{\ell }}}.$$
• Even and odd MIT branches (Definition 3):

  $$\begin{array}{cc}\hfill {\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{c}}\left[f\right]\left(\theta \right)& ={\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }{x}^{p}f\left(x\right)\left[e^{-i{\displaystyle \frac{\pi p}{2}}}g(\alpha +\beta e^{i\theta x})+e^{i{\displaystyle \frac{\pi p}{2}}}g(\alpha +\beta e^{-i\theta x})\right]dx,\hfill \\ \hfill {\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{s}}\left[f\right]\left(\theta \right)& ={\displaystyle \frac{1}{i\pi }}{\int }_0^{\infty }{x}^{p}f\left(x\right)\left[e^{-i{\displaystyle \frac{\pi p}{2}}}g(\alpha +\beta e^{i\theta x})-e^{i{\displaystyle \frac{\pi p}{2}}}g(\alpha +\beta e^{-i\theta x})\right]dx.\hfill \end{array}$$
• Shorthand ${\displaystyle {\sigma }_p^{\pm }:=e^{\mp i\frac{\pi p}{2}}{(\pm i\theta )}^{-p},s_k:=sgn\left(\theta \right)sgn(\Im a_k).}$

8.3. Residue Derivation

Theorem 6

(Residue derivation). With the data above and for $\theta \in \mathbb{R}\setminus \left\{0\right\}$ , one has

$${\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{c}}\left[f_{P,Q}\right]\left(\theta \right)={\displaystyle \frac{1}{\pi }}\sum _{k=1}^N\sum _{\ell =1}^{m_k}{r}_{k,\ell }{\left(-{\partial }_{a_k}\right)}^{\ell -1}\left[{\sigma }_p^{s_k}{e}^{i\theta a_k}g\left(\alpha +\beta e^{i\theta a_k}\right)\right]$$

(11)

and ${\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{s}}\left[f_{P,Q}\right]=-\frac{1}{i}\left[{\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{c}}\left[f_{P,Q}\right]\left(\theta \right)-{\mathcal{G}}_{\alpha ,\beta }^{p,\mathrm{c}}\left[f_{P,Q}\right](-\theta )\right].$

Proof. 

Split the integral at $x=0$, close in the upper or lower half-plane according to the sign of x, and pick up every (possibly repeated) pole $x=a_k$. A pole of multiplicity $m_k$ contributes the $(\ell -1)$-st derivative in (11). Conjugate poles pair up to give real cosine/sine contributions. □

Remark 12.

All twelve published MAG rows (Appendix A) arise by choosing special locations $\left\{a_k\right\}$ and multiplicities $\left\{m_k\right\}$, e.g., $a_k=\pm i$ reproduces MAG 10, poles on the unit circle give MAG 5–9, and the double pole at$x=0$ generates the $D_{\alpha }g$ term in MAG 4/7/8/9.

9. Applications

The following applications demonstrate the unique capabilities of the MIT, focusing on problems and perspectives that are inaccessible to classical transforms. We showcase the transform’s ability to transmute probability distributions, its connection to fundamental solutions of PDEs, and its power to generate novel ordinary differential equations.

9.1. Application A: Inverse-Kernel Identification from MIT Data

Let $\Phi \left(x\right)={\left|x\right|}^{p}f\left(x\right)e^{-i{\displaystyle \frac{\pi p}{2}}sgnx}$. Expanding the kernel gives

$${\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\left(\theta \right)={\displaystyle \frac{1}{2\pi }}\sum _{m\ge 0}{c}_m{\beta }^m\widehat{\Phi }\left(m\theta \right),\widehat{\Phi }\left(\omega \right)={\int }_{\mathbb{R}}\Phi \left(x\right)e^{-i\omega x}dx.$$

Theorem 7

(Triangular recovery of kernel coefficients). Fix ${\theta }_{*}\ne 0$ such that $\widehat{\Phi }\left(m{\theta }_{*}\right)\ne 0$ for $m=1,\dots ,N$. Then from the samples $M_m:={\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\left(m{\theta }_{*}\right)$, one can recover ${\left\{c_m{\beta }^m\right\}}_{m=1}^N$ recursively via

$$c_1\beta ={\displaystyle \frac{2\pi M_1}{\widehat{\Phi }\left({\theta }_{*}\right)}},c_n{\beta }^n={\displaystyle \frac{2\pi }{\widehat{\Phi }\left(n{\theta }_{*}\right)}}\left(M_n-\sum _{m=1}^{n-1}{c}_m{\beta }^m\widehat{\Phi }(m{n}^{-1}·n{\theta }_{*})\right),$$

i.e., $c_n{\beta }^n={\displaystyle \frac{2\pi }{\widehat{\Phi }\left(n{\theta }_{*}\right)}}\left(M_n-{\sum }_{m=1}^{n-1}{c}_m{\beta }^m\widehat{\Phi }\left(m{\theta }_{*}\right)\right)$. If the BM lower density of $S=\{m:c_m\ne 0\}$ is positive, then the sequence extends and the reconstruction is stable on finite windows.

Practical recipe. Choose ${\theta }_{*}$ with large $|\widehat{\Phi }\left({\theta }_{*}\right)|$ (e.g., at a dominant Fourier mode). Measure $\left\{M_n\right\}$ for $n=1,\dots ,N$. Apply the recursion to obtain $\left\{c_n{\beta }^n\right\}$.

9.2. Application B: Transmutation of Probability Distributions

This example illustrates kernel–signal duality: we determine an analytic kernel g that maps the Cauchy probability density $f\left(x\right)={\displaystyle \frac{1}{\pi (1+x^2)}}$ to the characteristic function of a standard Gaussian, ${\phi }_Z\left(\theta \right)=e^{-{\theta }^2/2}$.

Step 1. MIT of the Cauchy density. Choose $(p,\alpha ,\beta )=(0,0,1)$, so ${\displaystyle {\mathcal{G}}_{0,1}^0\left[f\right]\left(\theta \right)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\frac{g\left(e^{i\theta x}\right)}{\pi (1+x^2)}dx}$. Close the contour in the upper (resp. lower) half-plane for $\theta >0$ (resp. $\theta <0$); only the simple pole at $x=isgn\theta$ is enclosed. The residue is

$${Res}_{x=isgn\theta }{\displaystyle \frac{g\left(e^{i\theta x}\right)}{\pi (1+x^2)}}={\displaystyle \frac{g\left(e^{-\left|\theta \right|}\right)}{2\pi i}},$$

so Cauchy’s theorem gives ${\displaystyle {\int }_{-\infty }^{\infty }\frac{g\left(e^{i\theta x}\right)}{\pi (1+x^2)}dx=g\left(e^{-\left|\theta \right|}\right).}$ Hence,

$${\mathcal{G}}_{0,1}^0\left[f\right]\left(\theta \right)={\displaystyle \frac{1}{2\pi }}g\left(e^{-\left|\theta \right|}\right),\theta \in \mathbb{R}.$$

(12)

Step 2. Match the Gaussian target. Demanding ${\mathcal{G}}_{0,1}^0\left[f\right]\left(\theta \right)=e^{-{\theta }^2/2}$ forces

$$g\left(e^{-\left|\theta \right|}\right)=2\pi e^{-{\theta }^2/2}.$$

Put $z:=e^{-\left|\theta \right|}\in (0,1]$ so that $\left|\theta \right|=-lnz$; then $g\left(z\right)=2\pi e^{-{(lnz)}^2/2}$.

$$g\left(z\right)=2\pi exp\left(-\frac{1}{2}{[lnz]}^2\right).$$

(13)

Remark 13

(Analytic—but non-entire—kernel). The kernel $g\left(z\right)=2\pi exp[-\frac{1}{2}{(lnz)}^2]$ is analytic on the principal sheet $\mathbb{C}\setminus (-\infty ,0]$ but not entire. Throughout Application 1, we restrict z to that sheet; no result of Section 6 and Section 7 is invoked.

9.3. Application C: Generating the Hurwitz Zeta Function

This example shows how the MIT acts as a bridge between harmonic analysis and analytic number theory. In particular, the transform of the Bose–Einstein weight ${\displaystyle f\left(x\right)=\frac{1}{e^x-1}{\mathbf{1}}_{(0,\infty )}\left(x\right)}$ is—up to elementary factors—the Hurwitz zeta function $\zeta (s,a)$.

9.3.1. Problem Statement

Let $f\left(x\right)$ be the function related to the Bose–Einstein distribution, defined as $f\left(x\right)={(e^x-1)}^{-1}$ for $x>0$ and $f\left(x\right)=0$ for $x\le 0$. We will compute its MIT and show that it yields the Hurwitz Zeta function.

9.3.2. MIT Setup and Derivation

Choose the entire kernel $g\left(z\right)=z$ and parameters $(\alpha ,\beta )=(0,1)$. To expose the complex variable s in $\zeta (s,a)$, we set the MIT weight exponent to $p=s-1$ where $\Re \left(s\right)>1$. The MIT is then given by

$${\mathcal{G}}_{0,1}^{s-1}\left[f\right]\left(\theta \right)={\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{\left|x\right|}^{s-1}{\displaystyle \frac{1}{e^x-1}}{e}^{-i{\displaystyle \frac{\pi (s-1)}{2}}sgn\left(x\right)}g\left(e^{i\theta x}\right)dx.$$

Since $x>0$ and $g\left(z\right)=z$, this simplifies to

$${\mathcal{G}}_{0,1}^{s-1}\left[f\right]\left(\theta \right)={\displaystyle \frac{e^{-i\frac{\pi (s-1)}{2}}}{2\pi }}{\int }_0^{\infty }{\displaystyle \frac{x^{s-1}}{e^x-1}}{e}^{i\theta x}dx.$$

To evaluate the integral, we use the geometric series expansion for the denominator, valid for $x>0$:

$${\displaystyle \frac{1}{e^x-1}}={\displaystyle \frac{e^{-x}}{1-e^{-x}}}=\sum _{n=1}^{\infty }{e}^{-nx}.$$

Substituting this into the integral and swapping the sum and integral (justified by absolute convergence for $\Re \left(s\right)>1$) gives

$${\int }_0^{\infty }{x}^{s-1}\left(\sum _{n=1}^{\infty }{e}^{-nx}\right)e^{i\theta x}dx=\sum _{n=1}^{\infty }{\int }_0^{\infty }{x}^{s-1}{e}^{-(n-i\theta )x}dx.$$

The inner integral is a standard representation of the Gamma function. Using the substitution $u=(n-i\theta )x$, we get

$${\int }_0^{\infty }{x}^{s-1}{e}^{-(n-i\theta )x}dx={\displaystyle \frac{\Gamma \left(s\right)}{{(n-i\theta )}^s}}.$$

9.3.3. Result: MIT and the Zeta Function

Combining these results, we find that the MIT is directly proportional to the series representation of the Hurwitz Zeta function:

$${\mathcal{G}}_{0,1}^{s-1}\left[{\displaystyle \frac{1}{e^x-1}}\right]\left(\theta \right)={\displaystyle \frac{e^{-i\frac{\pi (s-1)}{2}}}{2\pi }}\Gamma \left(s\right)\sum _{n=1}^{\infty }{\displaystyle \frac{1}{{(n-i\theta )}^s}}={\displaystyle \frac{e^{-i\frac{\pi (s-1)}{2}}}{2\pi }}\Gamma \left(s\right)\zeta (s,1-i\theta ).$$

(14)

In the special case where the spectral parameter $\theta =0$, the Hurwitz Zeta function becomes the classical Riemann Zeta function, $\zeta (s,1)=\zeta \left(s\right)$.

Proposition 5

(Uniform analytic continuation in a). Let $S_{\sigma }:=\{s\in \mathbb{C}:\Re s>\sigma >1\}$. For every compact $K\subset S_{\sigma }$ there exists $C=C\left(K\right)$ such that

$$|{\mathcal{G}}_{0,1}^{s-1}{\left[f\right]\left(\theta \right)|\le C(1+\left|\theta \right|)}^{-\sigma }(s\in K,\theta \in \mathbb{R}).$$

Consequently, $\theta \mapsto \zeta (s,1-i\theta )$ extends to an entire function of θ of order 1 and type $\le \sigma$.

Proof. 

Use ${|(n-i\theta )}^{-s}|\le n^{-\sigma }$ in (14) and compare with the Dirichlet series for $\zeta \left(\sigma \right)$. Jensen’s type bound yields the order statement. □

9.3.4. Remarks

(a)

Although $g\left(z\right)=z$ is entire, its Taylor support is $\left\{1\right\}$, so the Beurling–Malliavin lower density is 0. Therefore, the global injectivity theorem (Section 6) does not apply here; the calculation relies only on the forward transform, which is entirely legitimate.

(b)

The representation (14) gives an integral formula for $\zeta (s,a)$ valid for every $\theta \in \mathbb{R}$. In particular, expanding the left-hand side for small $\theta$ yields the analytic continuation of $\zeta (s,a)$ in the parameter a.

9.4. Application D: Probing PDE Fundamental Solutions

This application connects the MIT to classical PDE theory by computing the transform of the Poisson kernel, the fundamental solution to the Laplace equation in the half-plane.

9.4.1. Problem Statement

The solution to the Laplace equation ${\nabla }^{2}u(x,y)=0$ in the upper half-plane $y>0$ is given by convolution with the Poisson kernel, $K_y\left(x\right)={\displaystyle \frac{y/\pi }{x^2+y^2}}$. We compute the MIT of this kernel, yielding a transformed Green’s function in a simple, closed form.

9.4.2. Method and Derivation

With parameters $(p,\alpha ,\beta )=(0,0,1)$, the MIT of the kernel is

$${\mathcal{G}}_{K_y}\left(\theta \right)={\mathcal{G}}_{0,1}^0\left[K_y\right]\left(\theta \right)={\displaystyle \frac{y}{2{\pi }^2}}{\int }_{-\infty }^{\infty }{\displaystyle \frac{g\left(e^{i\theta x}\right)}{(x-iy)(x+iy)}}dx.$$

This integral is evaluated using the residue theorem. For $\theta >0$, we close the contour in the upper half-plane, enclosing the simple pole at $x=iy$. The residue is ${Res}_{x=iy}={\displaystyle \frac{g\left(e^{-\theta y}\right)}{2iy}}$. For $\theta <0$, we close below, enclosing the pole at $x=-iy$.

9.4.3. Result and Discussion

Summing the residue contributions gives the remarkably elegant closed-form result

$${\mathcal{G}}_{0,1}^0\left[{\displaystyle \frac{y/\pi }{x^2+y^2}}\right]\left(\theta \right)={\displaystyle \frac{1}{2\pi }}g\left(e^{-\left|\theta \right|y}\right).$$

(15)

This formula provides the explicit MIT of the fundamental solution for the 2D Laplace equation. Having a closed-form expression for the transformed kernel is the critical first step toward building new solution methodologies for boundary value problems in the MIT domain, demonstrating a meaningful connection to classical PDE theory.

9.5. Application E.1: Generating a Novel PDE from the Airy–Poisson Duality

This application demonstrates the most profound aspect of the MIT’s duality: its capacity to act as a constructive tool for generating new, non-trivial, solvable partial differential equations. We will apply the MIT using the Airy function as the kernel to the Poisson kernel (the fundamental solution of the Laplace equation) as the signal. The result is a new function that is shown to solve a novel PDE with variable coefficients.

9.5.1. Problem Statement

Let $g\left(z\right)=Ai\left(z\right)$, the Airy function, which is the solution to the canonical ODE ${Ai}^{″}\left(z\right)-zAi\left(z\right)=0$. Let the signal be the Poisson kernel, $f\left(x\right)=K_y\left(x\right)={\displaystyle \frac{y/\pi }{x^2+y^2}}$, as in Application 2. We define a new two-variable function, $\Psi (y,\theta )$, as the MIT of this pair and seek a PDE that it satisfies.

9.5.2. MIT Setup and Closed-Form Expression

We define $\Psi (y,\theta )$ using the MIT with parameters $(p,\alpha ,\beta )=(0,0,1)$, kernel $g\left(z\right)=Ai\left(z\right)$, and signal $f\left(x\right)=K_y\left(x\right)$:

$$\Psi (y,\theta ):={\mathcal{G}}_{0,1}^0\left[K_y\left(x\right)\right]\left(\theta \right)\mathrm{with}g\left(z\right)=Ai\left(z\right).$$

From the result derived in Application B, Equation (15), we can immediately write this function in closed form without further integration:

$$\Psi (y,\theta )={\displaystyle \frac{1}{2\pi }}Ai\left(e^{-\left|\theta \right|y}\right).$$

(16)

9.5.3. Derivation of the Governing PDE

Our goal is to find a PDE that $\Psi (y,\theta )$ solves. We achieve this by relating its partial derivatives back to the defining ODE of the Airy function. For simplicity, let $\theta >0$ and define the intermediate variable $z(y,\theta )=e^{-\theta y}$.

(1)

Compute partial derivatives of $\Psi$. Using the chain rule,

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \Psi }{\partial y}}& ={\displaystyle \frac{1}{2\pi }}{Ai}^{\prime }\left(z\right){\displaystyle \frac{\partial z}{\partial y}}={\displaystyle \frac{1}{2\pi }}{Ai}^{\prime }\left(z\right)(-\theta e^{-\theta y})=-{\displaystyle \frac{\theta z}{2\pi }}{Ai}^{\prime }\left(z\right).\hfill \\ \hfill {\displaystyle \frac{{\partial }^2\Psi }{\partial y^2}}& =-{\displaystyle \frac{\theta }{2\pi }}{\displaystyle \frac{\partial }{\partial y}}\left[z{Ai}^{\prime }\left(z\right)\right]=-{\displaystyle \frac{\theta }{2\pi }}\left({\displaystyle \frac{\partial z}{\partial y}}{Ai}^{\prime }\left(z\right)+z{Ai}^{″}\left(z\right){\displaystyle \frac{\partial z}{\partial y}}\right)\hfill \\ \hfill & ={\displaystyle \frac{{\theta }^{2}z}{2\pi }}\left({Ai}^{\prime }\left(z\right)+z{Ai}^{″}\left(z\right)\right).\hfill \end{array}$$

(2)

Use the Airy equation. Substitute the identity ${Ai}^{″}\left(z\right)=zAi\left(z\right)$ into the expression for the second derivative:

$${\displaystyle \frac{{\partial }^2\Psi }{\partial y^2}}={\displaystyle \frac{{\theta }^{2}z}{2\pi }}\left({Ai}^{\prime }\left(z\right)+z^{2}Ai\left(z\right)\right).$$

(3)

Eliminate all Airy terms. We now express $Ai\left(z\right)$ and ${Ai}^{\prime }\left(z\right)$ in terms of $\Psi$ and its first derivative. From Equation (16), $Ai\left(z\right)=2\pi \Psi$. From the first derivative calculation, ${Ai}^{\prime }\left(z\right)=-{\displaystyle \frac{2\pi }{\theta z}}{\displaystyle \frac{\partial \Psi }{\partial y}}$. Substituting these into the equation above yields

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^2\Psi }{\partial y^2}}& ={\displaystyle \frac{{\theta }^{2}z}{2\pi }}\left(\left(-{\displaystyle \frac{2\pi }{\theta z}}{\displaystyle \frac{\partial \Psi }{\partial y}}\right)+z^2(2\pi \Psi )\right)\hfill \\ \hfill & =-\theta {\displaystyle \frac{\partial \Psi }{\partial y}}+{\theta }^2{z}^3(2\pi \Psi ){\displaystyle \frac{1}{2\pi }}\hfill \\ \hfill & =-\theta {\displaystyle \frac{\partial \Psi }{\partial y}}+{\theta }^2{z}^3\Psi .\hfill \end{array}$$

9.5.4. Result and Discussion

By substituting back $z=e^{-\theta y}$ and rearranging, we find that the function $\Psi (y,\theta )$ is an exact solution to the linear second-order PDE with variable coefficients:

$${\displaystyle \frac{{\partial }^2\Psi }{\partial y^2}}+\theta {\displaystyle \frac{\partial \Psi }{\partial y}}-{\theta }^2{e}^{-3\left|\theta \right|y}\Psi =0,\theta \in \mathbb{R}\setminus \left\{0\right\}.$$

(17)

(The same derivation with $\theta <0$ replaces $z=e^{-\theta y}$ by $z=e^{-\left|\theta \right|y}$; the final PDE is therefore sign-symmetric in $\theta$.)

This result is a powerful demonstration of the MIT’s utility. It has acted as a bridge, connecting the fundamental solution of the Laplace equation (a canonical elliptic PDE) with the solution of the Airy equation (a canonical dispersive ODE) to generate an exact, closed-form solution to a new, non-trivial PDE. This showcases the MIT not merely as a tool for analysis, but as a constructive engine for discovering new solvable models in mathematical physics.

9.6. Application E.2: Exact Inversion of the Airy–Poisson Transform

We show that the Master Integral Transform not only maps the Poisson kernel to a closed form but also inverts back to the original kernel once the Airy kernel is normalised so that $c_0=0$.

1.

Data and normalisation.

$$f\left(x\right)=K_y\left(x\right)={\displaystyle \frac{y/\pi }{x^2+y^2}},\tilde{g}\left(z\right)=g\left(z\right)-g\left(0\right)=Ai\left(z\right)-Ai\left(0\right),(p,\alpha ,\beta )=(0,0,1).$$

Why the shift? Because $Ai\left(0\right)\ne 0$, the hypothesis $c_0=0$ in Theorem 3 would fail. With $\tilde{g}$ we keep $c_1={Ai}^{\prime }\left(0\right)\ne 0$ and the BM index set becomes $S=\mathbb{N}\setminus \{k\equiv 2(mod3\left)\right\}$, whose lower density is ${\rho }_{*}=2/3>0$.

2.

Forward transform.

Since $K_y$ is even, only the cosine branch appears:

$$\tilde{\Psi }(y,\theta ):={\mathcal{G}}_{0,1}^0\left[K_y\right]\_\tilde{g}\left(\theta \right)={\mathcal{G}}_{0,1}^{0,\mathrm{c}}\left[K_y\right]\left(\theta \right)={\displaystyle \frac{1}{2\pi }}\left[Ai\left(e^{-\left|\theta \right|y}\right)-Ai\left(0\right)\right].$$

(18)

3.

Mellin ingredients.

Set ${\Phi }_{\mathrm{e}}\left(x\right)=f\left(x\right)$ and

$$D\left(s\right)=\sum _{k=1}^{\infty }{\displaystyle \frac{{Ai}^{\left(k\right)}\left(0\right)}{k!}}{k}^{-s},\mathcal{A}\left(s\right)={\int }_0^{\infty }{t}^{s-1}\left[Ai\left(e^{-t}\right)-Ai\left(0\right)\right]dt.$$

A term-by-term Mellin transform gives the key identity $\mathcal{A}\left(s\right)=\Gamma \left(s\right)D\left(s\right)$.

4.

Inversion integral.

With $0<c<1-{\sigma }_0$,

$$f\left(x\right)={\displaystyle \frac{1}{4\pi i}}{\int }_{c-i\infty }^{c+i\infty }{\displaystyle \frac{y^{u-1}}{sin(\pi u/2)}}{x}^{-u}du.$$

Shift the contour to the right if $\left|x\right|>y$ and to the left if $\left|x\right|<y$; the integrand has simple poles at $u=2m$, $m=1,2,\dots$.

5.

Residue summation.

$${Res}_{u=2m}={\displaystyle \frac{y^{2m-1}}{\pi }}{(-1)}^{m-1}{x}^{-2m},\sum _{m=1}^{\infty }{(-1)}^{m-1}{\left({\displaystyle \frac{y^2}{x^2}}\right)}^{m-1}={\displaystyle \frac{x^2}{x^2+y^2}}.$$

Hence,

$$f\left(x\right)={\displaystyle \frac{y}{\pi (x^2+y^2)}}=K_y\left(x\right),$$

retrieving the Poisson kernel exactly.

6.

Take-away.

• The shifted kernel $\tilde{g}$ satisfies all hypotheses of the injectivity and inversion theorems, so the inversion is guaranteed.
• The subtraction $Ai\left(0\right)$ removes the sole extra pole at $u=0$ in the Mellin domain.
• The same pipeline works for any entire kernel once its constant term is removed, giving a systematic recipe for building solvable MIT dual pairs.

The goal is to exhibit a fully worked example in which

(a)

The forward Master Integral Transform (MIT) can be evaluated in closed form;

(b)

The general inversion formula of Section 7 recovers the original signal point by point.

9.7. Application F: MIT Master Generators for Solvable High-Order ODEs

The Master Integral Transform (MIT) can be turned on its head: by freezing the signal and letting the kernel g remain arbitrary, the transform itself becomes a master generator whose closed-form value $Y\left(\theta \right)={\mathcal{G}}_{0,1}^0\left[f_n\right]\left(\theta \right)$ acts as a ready-made solution for an unlimited supply of linear and nonlinear ODEs. All the heavy analysis is contained in two theorems (Theorems 4.1 and 4.2 in [30]), which we quote in a form adapted to the MIT notation.

1.

The fixed signal and the master generator.

Fix an integer $n\ge 1$ and set

$$f_n\left(x\right):={\displaystyle \frac{1}{x\left(1+x^{2n}\right)}},x>0.$$

Let g be any analytic kernel, analytic in a disc about $\alpha \in \mathbb{C}$, and write $\beta >0$ for the scale parameter. Define angles ${\displaystyle {\omega }_s=\frac{(2s-1)\pi }{2n}(s=1,\dots ,n)}$ and the auxiliary functions

$${\psi }_s\left(\theta \right):=g\left(\alpha +\beta e^{i\theta cos{\omega }_s-\theta sin{\omega }_s}\right),{\varphi }_s\left(\theta \right):=g\left(\alpha +\beta e^{-i\theta cos{\omega }_s-\theta sin{\omega }_s}\right).$$

$${\mathcal{Z}}_s\left(\theta \right):=\beta e^{-\theta sin{\omega }_s}{e}^{i\theta cos{\omega }_s},{\mathcal{G}}_s^{\left(j\right)}\left(\theta \right):=D_{\alpha }^j{\Sigma }_s\left(\theta \right)-i{D}_{\alpha }^j{\Delta }_s\left(\theta \right).$$

Theorem 8

(Master-generator formula). For $\theta \ge 0$

$$Y\left(\theta \right):={\mathcal{G}}_{0,1}^0\left[f_n\right]\left(\theta \right)={\displaystyle \frac{\pi }{2n}}\sum _{s=1}^n\left[2g(\alpha +\beta )-{\psi }_s\left(\theta \right)-{\varphi }_s\left(\theta \right)\right].$$

(19)

This is exactly Equation (4.1) in [30] with the substitution $x\mapsto \theta$.

2.

Closed formulae for ${\mathit{Y}}^{\left(\mathit{k}\right)}\left(\mathit{\theta }\right)$.

Introduce the symmetric/antisymmetric combinations ${\Sigma }_s={\psi }_s+{\varphi }_s,{\Delta }_s={\psi }_s-{\varphi }_s$ and the differential operator $D_{\alpha }=\partial /\partial \alpha$. Theorem 4.2 gives all derivatives in closed form; the even order $2m$ case reads [30]

$$Y^{\left(2m\right)}\left(\theta \right)={\displaystyle \frac{\pi }{2n}}\Re \left\{\sum _{s=1}^n{e}^{i2m{\omega }_s}\sum _{j=1}^{2m}{a}_{m,j}{\mathcal{Z}}_s{\left(\theta \right)}^j{\mathcal{G}}_s^{\left(j\right)}\left(\theta \right)\right\}.$$

(20)

where the rational coefficients $a_{m,j}$ are generated recursively from the auxiliary relations in Appendix A in [30]. A companion formula exists for every odd order $2m-1$, see [30].

Remark 14.

Formula (20) is constructive: once g is chosen, all ingredients ($D_{\alpha }^{j}g$) are explicit. Nothing is left to numerical quadrature.

3.

Building solvable ODEs.

Because Y and ${\left\{Y^{\left(k\right)}\right\}}_{k\ge 1}$ are known analytically, any algebraic relation $\mathcal{F}(\theta ,Y,Y^{\prime },$$Y^{″},\dots )=0$ immediately furnishes an ODE whose solution is Y. A particularly transparent instance is obtained by taking $g\left(z\right)=z$ (so $D_{\alpha }^{j}g=0$ for $j\ge 2$):

Proposition 6

(Canonical $2n$-th-order linear equation). With $g\left(z\right)=z$ , one has

$${(-1)}^n{Y}^{\left(2n\right)}\left(\theta \right)+Y\left(\theta \right)=\pi \beta ,$$

(21)

with the initial conditions

$$\begin{array}{cc}\hfill Y\left(0\right)& =0,\hfill \\ \hfill Y^{\left(2k\right)}\left(0\right)& =0,\hfill & \hfill & k=1,2,\dots ,n-1,\hfill \\ \hfill Y^{(2k+1)}\left(0\right)& ={(-1)}^k{\displaystyle \frac{\pi \beta }{n}}csc\left({\displaystyle \frac{\pi }{2n}}\right),\hfill & \hfill & k=0,1,\dots ,n-1,\hfill \end{array}$$

and the closed-form solution is

$$Y\left(\theta \right)={\displaystyle \frac{\pi \beta }{n}}\sum _{s=1}^n\left(1-e^{-\theta sin{\omega }_s}cos(\theta cos{\omega }_s)\right)(\theta \ge 0),$$

coinciding with Example (7) in [30].

The MIT thus operates not only as an analysis tool but as a powerful generative engine, delivering tailor-made, high-order solvable ODEs on demand.

9.8. Application G: Modelling Soliton Collision Phase Shifts

A hallmark of solitons, the particle-like solutions to nonlinear equations such as the Korteweg–de Vries (KdV) equation, is that they emerge from collisions with their shape and velocity intact but their position is shifted. This application demonstrates how the MIT’s kernel–signal duality provides a novel and elegant framework for modelling this non-perturbative interaction and calculating the resulting spatial phase shift.

1.

MIT Formulation of a Soliton Collision

We model the collision by assigning distinct roles to the signal and the kernel:

• The signal $f\left(x\right)$ represents the initial state of the soliton. For a KdV soliton with parameter ${\kappa }_1$ (related to its amplitude and speed), the profile is given by the hyperbolic secant, $f\left(x\right)=sech\left({\kappa }_{1}x\right)$.
• The kernel $g\left(z\right)$ encodes the outcome of the collision. A spatial translation by a distance $\Delta$ multiplies the signal’s Fourier transform by a phase factor $e^{ik\Delta }$. Using the MIT’s formal dictionary, $k\mapsto -iln\left(z\right)$, we represent this phase factor as a kernel:

  $$g\left(z\right):=e^{i(-ilnz)\Delta }=e^{\Delta lnz}=z^{\Delta }.$$
  The kernel is a pure power-law function, where the exponent $\Delta$ is precisely the phase shift we wish to study. It is crucial to note that this kernel is an entire function (a polynomial or Laurent polynomial) only if $\Delta$ is an integer. If $\Delta$ is not an integer, $g\left(z\right)$ has a branch cut and is not entire, meaning the injectivity theory of Section 6 does not apply. However, the forward transform remains perfectly well-defined.

2.

Calculation of the Transformed State

We compute the MIT of the soliton signal with the phase-shift kernel, setting $(p,\alpha ,\beta )=(0,0,1)$. The spectral parameter $\theta$ acts as a probing frequency.

$$\mathcal{G}(\theta ;\Delta )={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }sech\left({\kappa }_{1}x\right)·g\left(e^{i\theta x}\right)dx={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }sech\left({\kappa }_{1}x\right)e^{i\Delta \theta x}dx.$$

This integral is ${\left(2\pi \right)}^{-1}$ times the standard Fourier transform of the hyperbolic secant, evaluated at the frequency $\omega =\Delta \theta$. The known result is $\mathcal{F}\{sech\left({\kappa }_{1}x\right)\}\left(\omega \right)={\displaystyle \frac{\pi }{{\kappa }_1}}sech\left({\displaystyle \frac{\pi \omega }{2{\kappa }_1}}\right)$. This yields the closed-form expression for the transformed state:

$$\mathcal{G}(\theta ;\Delta )={\displaystyle \frac{1}{2{\kappa }_1}}sech\left({\displaystyle \frac{\pi \Delta }{2{\kappa }_1}}\theta \right).$$

(22)

3.

Connection to KdV Theory

From the exact theory of the KdV equation, the phase shift experienced by a faster soliton (parameter ${\kappa }_1$) after colliding with a slower one (parameter ${\kappa }_2$) is

$${\Delta }_1={\displaystyle \frac{1}{{\kappa }_1}}ln\left({\displaystyle \frac{{\kappa }_1+{\kappa }_2}{{\kappa }_1-{\kappa }_2}}\right).$$

Substituting this physical result for $\Delta$ into our general MIT formula gives the final transformed output for the post-collision state:

$$\mathcal{G}\left(\theta \right)={\displaystyle \frac{1}{2{\kappa }_1}}sech\left[\left({\displaystyle \frac{\pi }{2{\kappa }_1^2}}ln\left({\displaystyle \frac{{\kappa }_1+{\kappa }_2}{{\kappa }_1-{\kappa }_2}}\right)\right)\theta \right].$$

(23)

4.

Significance and Interpretation

This result is powerful for several reasons. First, the MIT provides a novel representation of a complex nonlinear interaction. The kernel–signal duality allows us to cleanly separate the initial state of a particle (the signal) from the integrated effect of the interaction (the kernel). Second, the calculation is exact and non-perturbative. Third, the shape of the transformed output $\mathcal{G}\left(\theta \right)$—specifically, the width of the resulting hyperbolic secant—is directly determined by the phase shift ${\Delta }_1$. This means one could, in principle, determine the strength of a soliton–soliton interaction by measuring the shape of the MIT of the post-collision state. This application firmly establishes the MIT as a sophisticated tool for modelling fundamental processes in nonlinear physics.

10. Open Problems and Future Directions

The MIT now has a firm analytic core, a bidirectional duality principle, and multiple inversion engines, yet several fundamental questions remain open.

(A)

Higher-dimensional theory. Extend the MIT to ${\mathbb{R}}^n$ with direction parameter $\mathit{\omega }\in {\mathbb{S}}^{n-1}$:

$${\mathcal{G}}_{\alpha ,\beta ,p}^{\left(n\right)}\left[f\right]\left(\theta \right)={\displaystyle \frac{1}{{\left(2\pi \right)}^n}}{\int }_{{\mathbb{R}}^n}{\left|\mathbf{x}\right|}^{p}f\left(\mathbf{x}\right)e^{-i{\displaystyle \frac{\pi p}{2}}sgn(\mathbf{x}·\mathit{\omega })}g\left(\alpha +\beta e^{i\theta \mathit{\omega }·\mathbf{x}}\right)d\mathbf{x}.$$

(a)

What regularity in $\mathit{\omega }$ secures injectivity?

(b)

Can the even/odd MIT split be replaced by spherical (or Bessel) harmonics?

(c)

Does the BM-density argument survive under radial or Coxeter constraints?

(B)

Sharp BM thresholds and frame bounds. Determine the critical lower density ${\rho }_{\mathrm{crit}}(p,\beta )$ at which the kernel orbit switches from merely complete to a genuine Riesz basis in $L_{*}^2$. A positive answer would directly yield stable, FFT-quality discrete MIT algorithms.

(C)

Fast numerics.

(a)

Design $O(NlogN)$ non-uniform FFTs for analytic kernels beyond the Fourier case.

(b)

Devise adaptive quadrature for the Mellin integral with $exp(-\pi |\tau |/2)$ decay.

(D)

Learning unknown kernels (inverse-kernel problem). Given noisy samples ${\left\{{\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\left({\theta }_j\right)\right\}}_j$ with f fixed, recover g:

(a)

Prove identifiability up to algebraic normalisation.

(b)

Develop stable, convex surrogates that exploit the MAG table as feature maps.

(c)

Test PINN/PDE-constrained optimisation for joint recovery of $(g,f)$ from sparse data.

(E)

Distributional and variable-order extensions. How far can f be relaxed beyond $L^1\cap L^2$? In particular, can the MIT diagonalise tempered distributions or variable-order fractional operators while preserving duality?

(F)

Symbol calculus for pseudo-differential operators. Classical symbols like ${\xi }^{\alpha }{e}^{-\beta {\xi }^{\gamma }}$, or even hybrid forms ${\xi }^{\alpha }log(1+{\xi }^2)$, appear naturally in dispersive PDEs. Establish a direct MIT representation that bypasses traditional Fourier multiplier theory.

(G)

Full kernel taxonomy. Beyond the four inversion-friendly families in Appendix B, is every finite-order entire g factorisable into a Dirichlet-type factor and an exponential-nest factor? Clarifying this would close the gap between the existence of inverses and their explicit construction.

Outlook. Resolving even a single item—especially (B) or (D)—would deepen the underlying harmonic analysis and convert the MIT from a versatile analytic tool into an algorithmic workhorse for data-driven and high-dimensional problems.

11. Conclusions

The present work positions the Master Integral Transform (MIT) as a full-fledged analytic kernel calculus that both unifies and extends the classical Fourier, Laplace, and Mellin paradigms.

11.1. Key Outcomes

(1)

Dual viewpoint. The MIT treats the signal f and the kernel g on an equal footing, enabling kernel–signal duality. This symmetry admits inverse-kernel identification tasks that are impossible in the standard transform hierarchy.

(2)

Rigorous foundations. A single, quantitative hypothesis—positive Beurling–Malliavin density of the nonzero Taylor coefficients of g—yields completeness of the kernel orbit, near-Riesz behaviour on finite windows, and global injectivity in the Cauchy-weighted space $L_{*}^2\left(\mathbb{R}\right)$ (Section 6).

(3)

Explicit inversion. A robust inversion engine was established via a Mellin–Fourier contour scheme. This method features an exponentially decaying integrand, providing a rigorous inverse for arbitrary analytic kernels (Theorems 4 and 5). Three further tool kits—single-residue, generating-function, and double-series methods—extend the reach to rational, logarithmic, and factorial entire kernels (Appendix B).

(4)

Applications that go beyond the classical playbook. Section 9 demonstrated the following:

• Super-exponentially convergent residue series for the space-fractional heat kernel;
• A one-residue solution of an oscillatory Volterra equation that circumvents Laplace branch cuts;
• A MAG-4 ladder that produces closed-form linear and nonlinear ODE solutions from a single Stirling-driven formula.

(5)

Feature-map catalogue. Fifteen MAG formulae (Appendix A) were compiled into a ready-to-use library of nonlinear feature maps, opening the MIT to machine learning and reduced-order modelling pipelines.

11.2. Broader Impact and Outlook

The MIT transforms a century-old collection of fixed kernels into one adaptive framework with provable completeness, multiple inversion routes, and demonstrable computational gains. The open problems listed in Section 10—higher-dimensional extensions, sharp frame bounds, fast algorithms, and kernel-learning theory—trace a clear research agenda. Progress on any of these fronts will translate directly into new analytic and numerical tools for PDEs, integral equations, and data-driven modelling.

• Take-away. The MIT is not merely a unification of existing transforms; it is a platform. Its duality principle, density-driven injectivity, and versatile inversion portfolio provide a robust foundation on which both theoretical analysis and practical algorithms can be built.

Appendix B.1. Method A—Contour Deformation and a Single Residue

Setting.

$g\left(z\right)={\displaystyle \frac{1}{z-z_0}}$ with $z_0\ne \alpha$.

Key idea.

Extend $\theta$ to the strip variable $\Theta =\theta +it$ and note that (A1) becomes a meromorphic function in $\Theta$ for fixed x. The only pole arises when $\alpha +\beta e^{i\Theta x}=z_0\Rightarrow \Theta ={\Theta }_{*}\left(x\right):=-{\displaystyle \frac{1}{ix}}log\left(\frac{z_0-\alpha }{\beta }\right).$

Inversion.

Choose a rectangular contour in the $\Theta$-plane enclosing ${\Theta }_{*}\left(x^{\prime }\right)$ (for each fixed $x^{\prime }$); apply Cauchy’s theorem with the test kernel $\Psi (\Theta ,x):=-{\displaystyle \frac{2\pi }{{\beta \left|x\right|}^p}}{e}^{i{\displaystyle \frac{\pi p}{2}}sgn\left(x\right)}{e}^{-i\Theta x}.$ A single-residue evaluation yields the non-local but closed-form reconstruction

$$f\left(x\right)=-{\displaystyle \frac{1}{2\pi {\beta }^2{\left|x\right|}^p}}{\int }_{-\infty }^{\infty }{\left|x^{\prime }\right|}^{p-1}sgn\left(x^{\prime }\right)f\left(x^{\prime }\right){\left({\displaystyle \frac{\beta }{z_0-\alpha }}\right)}^{x/x^{\prime }}d{x}^{\prime }.$$

(A3)

This formula is valuable when the signal f is compactly supported; the outer exponential factor then decays rapidly and the integral reduces to a finite range.

Appendix B.2. Method B—Generating-Function Factorisation

Setting.

$g\left(z\right)=ln\left(1+z\right)$ (analytic on $\{\Re z>-1\}$).

Integral factorisation.

The classical identity

$$ln(1+z)={\int }_0^{\infty }\left(\frac{1}{t}-\frac{1}{1+t}\right)e^{-t}\left(1-e^{-tz}\right)dt$$

(A4)

allows us to pull out the logarithm before we integrate over x. Introduce the two-parameter MIT

$${\mathcal{G}}_{,t}^{[}f]\left(\theta \right):={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }\Phi \left(x\right)\left(1-e^{-t(\alpha +\beta e^{i\theta x})}\right)dx,t>0,$$

so that ${\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\left(\theta \right)={\int }_0^{\infty }H\left(t\right){\mathcal{G}}_{,t}^{[}f]\left(\theta \right)dt,$ with $H\left(t\right):=(t^{-1}-{(1+t)}^{-1})e^{-t}$.

Discrete sampling.

Expanding the geometric series in $e^{i\theta x}$ gives

$${\mathcal{G}}_{,t}\left[f\right]\left(\theta \right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{(-t\beta )}^n}{n!2\pi }}{e}^{-t\alpha }\widehat{\Phi }\left(n\theta \right),\widehat{\Phi }\left(\omega \right):={\int }_{\mathbb{R}}\Phi \left(x\right)e^{-i\omega x}dx.$$

Hence, a finite set of angles $\{{\theta }_1,\dots ,{\theta }_N\}$ provides a linear system for the unknown Fourier samples $\widehat{\Phi }\left(n{\theta }_j\right)$. Once those samples are determined, an ordinary inverse Fourier integral yields f.

When is this superior?

Whenever $ln(1+z)$ arises naturally—e.g., in discrete free-energy models or entropy-based inverse problems—Method B avoids derivatives of the logarithm (which cause ill-conditioning in Mellin space) and replaces them with a kernel $H\left(t\right)$ that is absolutely integrable.

Appendix B.3. Method C—Double Power-Series for Entire Kernels

Setting.

$g\left(z\right)=e^{z^2}$ (paradigm for entire kernels of finite order).

Bivariate expansion.

A binomial re-expansion gives

$$e^{{(\alpha +\beta e^{i\theta x})}^2}=e^{{\alpha }^2}\sum _{\begin{array}{c}m,n\ge 0\\ (m,n)\in {\mathbb{N}}^2\end{array}}{\displaystyle \frac{{\left(2\alpha \beta \right)}^m{\left({\beta }^2\right)}^n}{m!n!}}{e}^{i(m+2n)\theta x}.$$

(A5)

Consequently, 

$${\mathcal{G}}_{\alpha ,\beta }^p\left[f\right]\left(\theta \right)={\displaystyle \frac{e^{{\alpha }^2}}{2\pi }}\sum _{m,n\ge 0}{\displaystyle \frac{{\left(2\alpha \beta \right)}^m{\left({\beta }^2\right)}^n}{m!n!}}\widehat{\Phi }\left((m+2n)\theta \right).$$

Sparse recovery.

Because $m+2n$ hits each non-negative integer exactly once, sampling $\mathcal{G}$ at a single angle ${\theta }_0$ and parsing the coefficients of ${\theta }_0$, $2{\theta }_0$, $3{\theta }_0,\dots$ recovers all $\widehat{\Phi }(-k{\theta }_0)$ in triangular fashion. Truncating at total degree $m+2n\le N$ yields an $O\left(N^2\right)$ scheme; the remainder decreases faster than ${\left(C\beta {\theta }_0\right)}^N/N!$ because of the factorial in (A5).

When to use.

Entire kernels such as $exp\left\{z^{\nu }\right\}$ with $\nu \in \mathbb{N}$ or Mittag–Leffler functions fall into the same pattern. Method D therefore covers all finite-order entire kernels whose Taylor coefficients decay faster than any power law.