Conditional Exponential Convex Functions on White Noise Spaces

Abstract

This paper seeks to present the fundamental features of the category of conditional exponential convex functions (CECFs). Additionally, the study of continuous CECFs contributes to the characterization of convolution semigroups. In this context, we expand our focus to include a much broader class of Gaussian processes, where we define the generalized Fourier transform in a more straightforward manner. This approach is closely connected to the method by which we derived the Gaussian process, utilizing the framework of a Gelfand triple and the theorem of Bochner–Minlos. A part of this work involves constructing the reproducing kernel Hilbert spaces (RKHS) directly from CECFs.

1. Introduction

In this paper, the main traits within the class of CECFs, defined on certain types of white noise spaces, are presented. Let Q represent a locally compact basis on ${\mathbb{R}}^d$ [1]. Let $C_0\left(Q\right)$ be the set consisting of all continuous functions with complex values. The space $C_b\left(Q\right)$, which consists of continuous complex-valued bounded functions, is a complete normed space with respect to the given norm.

$${\parallel f\parallel }_{\infty }=\underset{x\in Q}{sup}\left|f\left(x\right)\right|,$$

(1)

where f is defined on Q. The space of functions on Q that are infinitely differentiable and bounded will be referred to as $C_b^{\infty }\left(Q\right)$, moreover, by $S\left(Q\right)$, the linear subspace of $C_b^{\infty }\left(Q\right)$ created by the set that contains functions on Q such as $x^{\alpha }{D}^{\beta }f\left(x\right)\le c_{\alpha ,\beta }$ with $\alpha ,\beta \in {\mathbb{Z}}_{+}^n$ and a constant $c_{\alpha ,\beta }$. The space of tempered distributions is represented by $S^{*}$, which is the dual of $S\left(Q\right)$. Numerous studies have focused on exploring spaces of white noise [2,3]. The papers focus on the development of test spaces, generalized functions, and operators acting within these spaces [1,4,5,6,7,8,9,10]. Distributions are critical in areas like partial differential equations (PDEs) and quantum field theory [11,12,13,14,15,16,17,18], where quantum fields are modeled as operator-valued distributions. The modern theory of generalized functions, particularly those involving infinitely many variables, has roots in the foundational works of Berezansky, Samoilenko [19], and Hida [20]. In [19], the authors introduced the idea of constructing test and generalized function spaces as infinite tensor products of one-dimensional spaces, a key advancement in the field. In [20], the established method for developing the theory of generalized functions was employed. However, the functions considered were viewed as dependent on a point in an infinite-dimensional space, where the Gaussian measure was introduced. This measure played a role analogous to that of the Lebesgue measure in the classical theory of generalized functions. In actuality, the classical approach to the construction of the theory of generalized functions was employed.

The goal of this paper is to introduce the key attributes of the class of CECFs, moreover, the analysis of continuous CECFs results in the characterization of convolution semigroups, where we broaden our focus to encompass a much larger family of Gaussian processes. This class of functions is particularly useful in stochastic calculus and white noise analysis, offering a framework for understanding the structure of semigroups arising in random processes.

The structure of this paper is as follows: Section 2 introduces the fundamental properties of the class of CECFs defined on the space of rabidly decreasing functions on Q. In Section 3, we introduce the Bochner–Minlos-type formula for CECFs and speak about the correspondence between convolution semigroups ${\left({\mu }_t\right)}_{t\ge 0}$ on S and the set of all continuous CECFs, defined on $S^{*}$. In Section 4, a new approach for constructing spaces of generalized functions is presented and accentuate that the RKHS is obtained directly from CECFs that we start from. In Section 5, we introduce the generalized Fourier transform on infinite spaces, and we find that the image of the real-valued function $T_{\lambda }$ is contained in RKHS ${\mathcal{A}}_{G_{\lambda }}$. In Section 6, we conclude with a summary and final observations.

2. White Noise Spaces Induced by Conditional Exponential Convex Functions

In [21], we examined the collection of strongly negative definite functions on product dual hypergroups and utilized their characteristics to provide a proof of the Lévy–Khinchin formula. In the present section, we will explore a certain characteristic of the class of CECFs and defined on a hypergroup. The study of continuous CECFs provides a foundation for characterizing convolution semigroups, as discussed in ([22], Section 2).This characterization plays a crucial role in potential theory. The elements of $S\left(Q\right)$ are referred to as rapidly decreasing functions, and for each $\alpha ,\beta \in {\mathbb{Z}}_{+}^n$, $S\left(Q\right)$ is endowed with the family of seminorms

$${\parallel f\parallel }_{\alpha ,\beta }=\underset{x\in Q}{sup}\left|x^{\alpha }{D}^{\beta }f\left(x\right)\right|.$$

(2)

A function $\psi :S^{*}\to \mathbb{C}$ is called conditionally exponential convex if it satisfies the following inequality for all $n\in \mathbb{N}$, any points $y_1,y_2,\dots ,y_n\in G^{*}\subseteq S^{*}$, and any coefficients $c_1,c_2,\dots ,c_n\in \mathbb{C}$:

$$\sum _{i,j=1}^n\left[\psi \left(y_i\right)+\psi \left(y_j\right)-\psi (y_i+y_j)\right]c_i\overline{c_j}\ge 0.$$

(3)

The inclusion of the summation over all pairs ensures that the function correctly models the necessary convexity behavior, reinforcing its role in structural analysis. This formulation helps in maintaining consistency across different functional spaces.

Let $G:Q\to \mathbb{C}$ be a continuous CECF if for all $n\in \mathbb{N}$, we define

$${⟨·,·⟩}_G:C_0\left(Q\right)\times C_0\left(Q\right)\to \mathbb{C}$$

(4)

by

$${⟨\phi ,\psi ⟩}_G:={\int }_Q{\int }_{Q}G(x-y)\phi \left(x\right)\overline{\psi \left(y\right)}d\mu \left(x\right)d\nu \left(y\right),$$

(5)

where $\mu ,\nu \in \mathcal{M}\left(Q\right)$ the Radon measure space on Q. The inner product ${⟨·,·⟩}_G$ satisfies the following conditions:

I.

${⟨·,·⟩}_G$ in the first coordinate is complex linear and in the second conjugate complex linear, i.e., for any $\phi ,\psi \in C_0\left(Q\right)$ and any $c\in \mathbb{C}$

$${⟨c\phi ,\psi ⟩}_G=c{⟨\phi ,\psi ⟩}_G\mathrm{and}{⟨\phi ,c\psi ⟩}_G=\overline{c}{⟨\phi ,\psi ⟩}_G.$$

II.

${⟨·,·⟩}_G$ is conjugate symmetric, i.e., for any $\phi ,\psi \in C_0\left(Q\right)$

$${⟨\phi ,\psi ⟩}_G=\overline{{⟨\psi ,\phi ⟩}_G}.$$

III.

${⟨·,·⟩}_G$ is positive definite, meaning that for any $\phi \in C_0\left(Q\right)$

$${⟨\phi ,\phi ⟩}_G={\int }_Q{\int }_{Q}G(x-y)\phi \left(x\right)\overline{\phi \left(y\right)}d\mu \left(x\right)d\nu \left(y\right)\ge 0.$$

Theorem 1.

For any continuous CECF G on Q, the inner product space $\left(C_0\left(Q\right),{⟨·,·⟩}_G\right)$ is a complex Hilbert space.

Proof. 

We have that ${⟨·,·⟩}_G$ is an inner product space and $C_0\left(Q\right)$ is an infinite space, so all we need to prove is the completeness of that space. We assume that we have a Cauchy sequence $\left\{{\phi }_n\right\}$, and we need to prove that this Cauchy sequence converges to a limit in $\left(C_0\left(Q\right),{⟨·,·⟩}_G\right)$. We have,

$${⟨\phi ,\psi ⟩}_G={\int }_Q{\int }_{Q}G(x-y)\phi \left(x\right)\overline{\psi \left(y\right)}d\mu \left(x\right)d\nu \left(y\right),$$

where $\phi ,\psi \in C_0\left(Q\right)$ and $\mu ,\nu \in \mathcal{M}\left(Q\right)$.

$$\begin{array}{ccc}\hfill \parallel {\phi }_n-{\phi }_m{\parallel }^2& =& {⟨{\phi }_n-{\phi }_m,{\phi }_n-{\phi }_m⟩}_G\hfill \\ & =& {\int }_Q{\int }_{Q}G(x-y)({\phi }_n-{\phi }_m)\left(x\right)({\phi }_n-{\phi }_m)\left(y\right)d\mu \left(x\right)d\nu \left(y\right)\hfill \\ & \le & {\int }_Q{\int }_Q{\displaystyle \frac{\left|G\right(x-y\left)\right|}{2}}d\nu \left(y\right){|{\phi }_n\left(x\right)-{\phi }_m\left(x\right)|}^{2}d\mu \left(x\right)\hfill \\ & +& {\int }_Q{\int }_Q{\displaystyle \frac{\left|G\right(x-y\left)\right|}{2}}d\mu \left(x\right){|{\phi }_n\left(y\right)-{\phi }_m\left(y\right)|}^{2}d\nu \left(y\right)\hfill \\ & \to & 0\mathrm{as}n,m\to \infty .\hfill \end{array}$$

(6)

Hence,

$$|{\phi }_n\left(x\right)-{\phi }_m\left(x\right)|,|{\phi }_n\left(y\right)-{\phi }_m\left(y\right)|\to 0\mathrm{as}n,m\to \infty .$$

Since $\left\{{\phi }_n\right\}$ is a Cauchy sequence and we have that $C_0\left(Q\right)$ is a complete space, which means that $\underset{n\to \infty }{lim}{\phi }_n=\phi$ as $n\to \infty$, i.e.,

$$|{\phi }_n\left(x\right)-\phi \left(x\right)|\to 0\mathrm{as}n\to \infty ,$$

which tends to that $\phi$ belonging to $\left(C_0\left(Q\right),{⟨·,·⟩}_G\right)$, so this space is complete. □

Corollary 1.

For any continuous CECF G on Q, the space ${\mathcal{H}}_G:=(C_0\left(Q\right),{⟨·,·⟩}_G)$ is a subspace of Hilbert space $L^2\left(\mu \right)$.

Proof. 

We want to prove ${\mathcal{H}}_G\subset L^2\left(\mu \right)$, so let $\phi ,\psi \in {\mathcal{H}}_G$, and we want to prove that $\phi ,\psi \in L^2\left(\mu \right)$. Assume that

$${\int }_Q\left|G(x-y)\right|d\mu \left(x\right)\le R_1,\mathrm{for}\mathrm{all}y\in Q$$

and

$${\int }_Q\left|G(x-y)\right|d\nu \left(y\right)\le R_2,\mathrm{for}\mathrm{all}x\in Q,$$

where $R_1$ and $R_2$ are positive constants, and by using (5).

$$\left|{⟨\phi ,\psi ⟩}_G\right|=\left|{\int }_Q{\int }_{Q}G(x-y)\phi \left(x\right)\overline{\psi \left(y\right)}d\mu \left(x\right)d\nu \left(y\right)\right|,$$

(7)

where by using a Cauchy–Young inequality [1], we have

$$\begin{array}{ccc}\hfill \left|\phi \left(x\right)\overline{\psi \left(y\right)}\right|& \le & {\displaystyle \frac{{\left|\phi \left(x\right)\right|}^2}{2}}+{\displaystyle \frac{{\left|\psi \left(y\right)\right|}^2}{2}}\hfill \\ & \le & {\int }_Q{\int }_Q{\displaystyle \frac{\left|G\right(x-y\left)\right|}{2}}d\nu \left(y\right){\left|\phi \left(x\right)\right|}^{2}d\mu \left(x\right)\hfill \\ & +& {\int }_Q{\int }_Q{\displaystyle \frac{\left|G\right(x-y\left)\right|}{2}}d\mu \left(x\right){\left|\psi \left(y\right)\right|}^{2}d\nu \left(y\right)\hfill \\ & \le & {\displaystyle \frac{R_2}{2}}{\parallel \phi \parallel }_{L^2\left(\mu \right)}^2+{\displaystyle \frac{R_1}{2}}{\parallel \psi \parallel }_{L^2\left(\nu \right)}^2.\hfill \end{array}$$

(8)

So, $\psi \in L^2\left(\mu \right)$. □

Let ${\mathcal{M}}_c$ represent the set of all continuously real-valued functions $\omega$ on ${\mathbb{R}}^n$ that fulfill the requirements listed below:

(1)

$0=\omega \left(0\right)\le \omega (\zeta +\eta )\le \omega \left(\zeta \right)+\omega \left(\eta \right);\zeta ,\eta \in {\mathbb{R}}^n$.

(2)

${\int }_{{\mathbb{R}}^n}{\displaystyle \frac{\omega \left(\zeta \right)}{{(1+|\zeta \left|\right)}^{n+1}}}d\zeta <\infty$.

(3)

$\omega \left(\zeta \right)\ge a+blog(1+|\zeta \left|\right)$ for some constant $a,b$.

(4)

$\omega \left(\zeta \right)$ is radial.

With the weight function $\omega$ in ${\mathcal{M}}_c$ and open set $\Omega \in {\mathbb{R}}^n$, Björck extends the Schwartz space by the space $S_{\omega }$ of all $C^{\infty }$-function $\phi \in L^1\left({\mathbb{R}}^n\right)$:

$$P_{\alpha ,\lambda }\left(\phi \right)=\underset{x\in {\mathbb{R}}^n}{sup}{e}^{\lambda \omega \left(x\right)}\left|D^{\alpha }\phi \left(x\right)\right|<\infty ,$$

(9)

and

$${\Pi }_{\alpha ,\lambda }\left(\phi \right)=\underset{\zeta \in {\mathbb{R}}^n}{sup}{e}^{\lambda \omega \left(\zeta \right)}\left|D^{\alpha }\widehat{\phi }\left(\zeta \right)\right|<\infty ,$$

(10)

and $S_{\omega }^{*}$ the dual space of $S_{\omega }$. Let g be a CECF and

$${\omega }_g\left(\zeta \right)=log(1+|g\left(\zeta \right)\left|\right).$$

(11)

For $s\in \mathbb{R}$, we define by ${\mathcal{H}}_{\omega ,g}^s$ as the collection of all generalized distributions $u\in S_{\omega }^{*}$:

$${\parallel u\parallel }_g^{\omega ,s}={\left({\int }_{{\mathbb{R}}^n}{e}^{2s{\omega }_g\left(\zeta \right)}{\left|\widehat{u}\left(\zeta \right)\right|}^{2}d\zeta \right)}^{{\displaystyle \frac{1}{2}}}.$$

(12)

Theorem 2.

The space ${\mathcal{H}}_g^{\omega ,s}$ is a Hilbert space with an inner product denoted by

$${⟨u,v⟩}_g^{\omega ,s}={\int }_{{\mathbb{R}}^n}{e}^{2s{\omega }_g\left(\zeta \right)}\widehat{u}\left(\zeta \right)\overline{\widehat{v}\left(\zeta \right)}d\zeta .$$

(13)

Proof. 

We want to prove that the space ${\mathcal{H}}_g^{\omega ,s}$ is complete, so we suppose that we have a Cauchy sequence $\left\{u_m\right\}$ in ${\mathcal{H}}_g^{\omega ,s}$, and we want to prove that this Cauchy converges to a limit u in ${\mathcal{H}}_g^{\omega ,s}$, where the norm is defined by:

$${\parallel u\parallel }_g^{\omega ,s}={\left({\int }_{{\mathbb{R}}^n}{e}^{2s{\omega }_g\left(\zeta \right)}{\left|\widehat{u}\left(\zeta \right)\right|}^{2}d\zeta \right)}^{{\displaystyle \frac{1}{2}}},$$

where

$${\parallel u\parallel }_g^{\omega ,s}={\parallel e^{s{\omega }_g\left(\zeta \right)}\widehat{u}\left(\zeta \right)\parallel }_{L^2}.$$

(14)

Therefore, $\left\{e^{s{\omega }_g\left(\zeta \right)}\widehat{u_m}\left(\zeta \right)\right\}$ forms a Cauchy sequence in $L_2$, and consequently, it converges to v in $L_2$. Assume that u is a tempered distribution, which we can write as $u={\mathcal{F}}^{-1}\left(e^{-s{\omega }_g\left(\zeta \right)}\widehat{v}\right)$. We have $\widehat{u}=e^{-s{\omega }_g\left(\zeta \right)}\widehat{v}$ and $v\in L_2$, then u in ${\mathcal{H}}_g^{\omega ,s}$, where

$$\parallel u_m{-u\parallel }_g^{\omega ,s}={\parallel e^{s{\omega }_g\left(\zeta \right)}\widehat{u_m}-v\parallel }_{L_2}\to 0\mathrm{as}m\to \infty ,$$

(15)

i.e., the Cauchy sequence $u_m$ converges to a limit u in ${\mathcal{H}}_g^{\omega ,s}$. □

Lemma 1.

Let $u\in {\mathcal{H}}_g^{\omega ,s}$, $<u,·{>}_g^{\omega ,s}$ be the conjugate linear functional on $S_{\omega }$, which uniquely extends to a conjugate linear functional on ${\mathcal{H}}_g^{\omega ,s}$ and satisfies the following:

(1) 

${⟨u,v⟩}_g^{\omega ,s}={\left(2\pi \right)}^{-n}{\int }_{{\mathbb{R}}^n}{e}^{2s{\omega }_g\left(\zeta \right)}\widehat{u}\left(\zeta \right)\overline{\widehat{v}\left(\zeta \right)}d\zeta$,

(2) 

${|⟨u,v⟩}_g^{\omega ,s}{|\le \parallel u\parallel }_g^{\omega ,s}{\parallel v\parallel }_g^{\omega ,s},u\in {\mathcal{H}}_g^{\omega ,s},v\in {\mathcal{H}}_g^{\omega ,-s}$,

(3) 

${⟨u,v⟩}_g^{\omega ,s}=\overline{{⟨v,u⟩}_g^{\omega ,s}}$.

Theorem 3.

The space $S_{\omega }$ is dense in ${\mathcal{H}}_g^{\omega ,s}$ for all $s\in \mathbb{R}$.

Proof. 

To prove that $S_{\omega }$ is dense in ${\mathcal{H}}_g^{\omega ,s}$, we need to check two things: the first is that $S_{\omega }\subset {\mathcal{H}}_g^{\omega ,s}$ and the second is that $S_{\omega }={\mathcal{H}}_g^{\omega ,s}$. For the first, let us have a bijective map $h_s:S_{\omega }\to S_{\omega }$, $v\mapsto e^{s{\omega }_g\left(\zeta \right)}\widehat{v}$. From (11) and the fact that g is a CECF, we have $e^{s{\omega }_g\left(\zeta \right)}\widehat{v}\in S_{\omega }\subset L_2$, which leads to $S_{\omega }\subset {\mathcal{H}}_g^{\omega ,s}$. Secondly, we prove that $\overline{S_{\omega }}={\mathcal{H}}_g^{\omega ,s}$, so we must prove that $S_{\omega }^{\perp }=\left\{0\right\}$, where

$$S_{\omega }^{\perp }=\{v\in {\mathcal{H}}_g^{\omega ,s}:{⟨v,\phi ⟩}_g^{\omega ,s}=0\forall \phi \in S_{\omega }\}.$$

(16)

We want to arrive at $v=0$, i.e., $v\in {\mathcal{H}}_g^{\omega ,s}$ leads to ${⟨v,\phi ⟩}_g^{\omega ,s}=0\forall \phi \in S_{\omega }$. We have

$${⟨v,\phi ⟩}_g^{\omega ,s}={⟨e^{s{\omega }_g\left(\zeta \right)}\widehat{v},e^{s{\omega }_g\left(\zeta \right)}\widehat{\phi }⟩}_{L_2}.$$

(17)

Since $h_s$ is bijective $\forall \phi \in S_{\omega }$, we find ${⟨e^{s{\omega }_g\left(\zeta \right)}\widehat{v},\widehat{\phi }⟩}_{L_2}=0$; since $S_{\omega }$ is dense in $L_p$, $1\le p<\infty$ [11], that means that $e^{s{\omega }_g\left(\zeta \right)}\widehat{v}=0$. So $v=0$, i.e., $S_{\omega }^{\perp }=\left\{0\right\}$. Therefore, $\overline{S_{\omega }}={\mathcal{H}}_g^{\omega ,s}$, which completes the proof. □

Corollary 2.

${\mathcal{H}}_g^{\omega ,t}\subset {\mathcal{H}}_g^{\omega ,s}$ for $t>s$; the inclusion is continuous and has a dense image.

Proof. 

The proof can be directly from Theorem 3. □

3. Bochner–Minlos-Type Formula for Conditional Exponential Convex Functions

The Bochner–Minlos-type formula plays a pivotal role in the study of CECFs. It generalizes key results from characteristic functions, establishing a profound link between convexity, Fourier transforms, and probability distributions.

Theorem 4.

A continuous function $\psi :S^{*}\to \mathbb{C}$ is conditionally exponential convex if the following requirements are fulfilled:

(I) 

$\psi \left(0\right)\ge 0$,

(ii) 

The function ${\Psi }_t\left(r\right)=exp[-t\psi \left(r\right)]$ is continuous and conditionally exponential convex for all $t>0$.

Proof. 

We need to prove $\left(ii\right)$, where it is clear that $\psi \left(0\right)\ge 0$. We have

$$\sum _{i,j=1}^n\left[\psi \left(r_i\right)+\overline{\psi \left(r_j\right)}-\psi (r_i+r_j)\right]c_i\overline{c_j}\ge 0.$$

(18)

So,

$$\sum _{i,j=1}^{n}exp\left[\psi \left(r_i\right)+\overline{\psi \left(r_j\right)}-\psi (r_i+r_j)\right]c_i\overline{c_j}\ge 0.$$

(19)

For $t=1$, we obtain

$$\begin{array}{ccc}\hfill \sum _{i,j=1}^n{\Psi }_1(r_i+r_j)c_i\overline{c_j}& =& \sum _{i,j=1}^{n}exp\left[-\psi (r_i+r_j)\right]c_i\overline{c_j}\hfill \\ & =& \sum _{i,j=1}^{n}exp\left[-\psi (r_i+r_j)-\psi \left(r_i\right)+\psi \left(r_i\right)+\overline{\psi \left(r_j\right)}-\overline{\psi \left(r_j\right)}\right]c_i\overline{c_j}\hfill \\ & =& \sum _{i,j=1}^{n}exp\left[\psi \left(r_i\right)+\overline{\psi \left(r_j\right)}-\psi (r_i+r_j)\right]c_{i}exp[-\psi \left(r_i\right)]\overline{c_j}exp[-\overline{\psi \left(r_j\right)}]\hfill \\ & =& \sum _{i,j=1}^{n}exp\left[\psi \left(r_i\right)+\overline{\psi \left(r_j\right)}-\psi (r_i+r_j)\right]{\alpha }_i\overline{{\alpha }_j},\hfill \end{array}$$

(20)

where $c_{i}exp[-\psi \left(r_i\right)]={\alpha }_i$ and $\overline{c_j}exp[-\overline{\psi \left(r_j\right)}]=\overline{{\alpha }_j}$. So, ${\Psi }_1$ is CECF, according to the mathematical induction, then ${\Psi }_t$ is CECF for all $t>0$. □

Corollary 3.

Let $\psi :S^{*}\to \mathbb{C}$ be a CECF and suppose that $\psi \left(0\right)\ge 0$, then ${\displaystyle \frac{1}{\psi }}$ is CECF.

Proof. 

Since $\psi$ is CECF, then $exp[-t\psi (r\left)\right]$ is CECF for all $t>0$.

$${\displaystyle \frac{1}{\psi }}={\int }_0^{\infty }exp[-t\psi \left(r\right)]dt.$$

(21)

Hence, we have

$$\begin{array}{ccc}\hfill \sum _{i,j=1}^n{\displaystyle \frac{1}{\psi (r_i+r_j)}}{c}_i\overline{c_j}& =& \sum _{i,j=1}^n{c}_i\overline{c_j}{\int }_0^{\infty }exp[-t\psi (r_i+r_j)]dt\hfill \\ & =& {\int }_0^{\infty }\sum _{i,j=1}^{n}exp[-t\psi (r_i+r_j)]c_i\overline{c_j}dt,\hfill \end{array}$$

which is clear from Theorem 4 that ${\displaystyle \frac{1}{\psi }}$ is CECF. □

Theorem 5.

A bijective relationship exists between convolution semigroups ${\left({\mu }_t\right)}_{t>0}$ on S and the set of continuous CECFs on $S^{*}$. In particular, for any convolution semigroup ${\left({\mu }_t\right)}_{t>0}$ on S, there is a unique continuous CECF ψ defined on $S^{*}$ that satisfies:

$$\widehat{{\mu }_t}\left(g\right)=exp[-t\psi \left(g\right)]\mathit{for}t>0\mathit{and}g\in S^{*}.$$

(22)

Here, $\widehat{{\mu }_t}$ refers to the Fourier transform of ${\mu }_t$. Conversely, if a continuous CECF is given on $S^{*}$, Equation (22) uniquely defines a convolution semigroup ${\left({\mu }_t\right)}_{t>0}$ on S.

Proof. 

Assume that ${\left({\mu }_t\right)}_{t>0}$ is a convolution semigroup on S; let the function ${\psi }_g\left(t\right):(0,\infty )\to \mathbb{C}$ be defined by ${\psi }_g\left(t\right)=\widehat{{\mu }_t}\left(g\right)$ for fixed $g\in S^{*}$ and $t>0$. Clearly, the continuous function ${\psi }_g$ satisfies the following:

$${\psi }_g(s+t)={\psi }_g\left(s\right){\psi }_g\left(t\right).$$

(23)

Also, the complex number $\phi \left(g\right)$ is uniquely determined in such a way that $\psi \left(0\right)\ge 0,{\psi }_g\left(t\right)=exp(-t\phi \left(g\right))\mathrm{for}t>0$ and $g\to exp(-t\phi \left(g\right))=\widehat{{\mu }_t}\left(g\right)$ is CECF and continuous for $t>0$. Then, from Theorem 4, we find that $\phi$ is CECF. The converse can easily be proved. □

Theorem 6.

(Main Result) If $g:S\left(Q\right)\to \mathbb{C}$ is continuous w.r.t. the Fréchet topology, $g\left(0\right)=1$ and CECF, then a unique probability measure P on $(S^{*}\left(Q\right),B\left(S^{*}\right))$ exists such that

$${\mathbb{E}}_P\left[exp[i⟨·,s⟩]\right]:={\int }_{s^{*}}exp\left[i⟨\omega ,s⟩\right]dP\left(\omega \right)=g\left(s\right),$$

(24)

for all $s\in S\left(Q\right)$.

4. Reproducing Kernel Hilbert Space ${\mathcal{A}}_G$

Reproducing kernel Hilbert spaces (RKHS) have emerged as a key mathematical framework across numerous disciplines, particularly in statistics and machine learning [23,24,25], where they are widely applied. This work develops a framework for constructing spaces of generalized functions using CECFs, where the RKHS ${\mathcal{A}}_G$ is directly derived from these functions.

Let G be a continuous CECF on ${\mathbb{R}}^d$, and set $G_y\left(\mathbf{x}\right):=G(\mathbf{x}-\mathbf{y})\mathrm{for}\mathrm{all}\mathbf{x},\mathbf{y}\in {\mathbb{R}}^d.$ Define:

$$(\phi \ast G)\left(\mathbf{x}\right):={\int }_{{\mathbb{R}}^d}\phi \left(\mathbf{y}\right)G_y\left(\mathbf{x}\right)d\mathbf{y},\phi \in S^d,$$

(25)

and

$${⟨\phi \ast G,\psi \ast G⟩}_{{\mathcal{A}}_G}:={\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}G(\mathbf{x}-\mathbf{y})\phi \left(\mathbf{x}\right)\overline{\psi \left(\mathbf{y}\right)}d\mathbf{x}d\mathbf{y},$$

(26)

for all $\phi ,\psi \in S^d$. Then, $\phi *G$, $\phi \in S^d$ forms a pre-Hilbert space ${\mathcal{A}}_o$ with the inner-product ${⟨·,·⟩}_{{\mathcal{A}}_G}$.

Theorem 7.

Let $k\in \mathbb{N}$ and $\Lambda \in D_k$, and set $G_{\Lambda }\left(x\right)={\sum }_{\lambda \in \Lambda }{e}^{i\lambda x}$, $x\in {\mathbb{R}}^k$ as a conditional exponential convex tempered distribution. Let ${\mathcal{A}}_{G_{\Lambda }}$ be the generalized RKHS of Schwartz.

Then a function h on ${\mathbb{R}}^k$ is in ${\mathcal{A}}_{G_{\Lambda }}$ if and only if it has a convolution factorization $h=\phi *G_{\Lambda }$, where φ is a measurable function such that $\widehat{\phi }\left(\lambda \right)$ exists for all $\lambda \in \Lambda$, and $\widehat{\phi }\left(\lambda \right)$, $\lambda \in \Lambda$ is in $L_2(\Lambda )$ and

$${\parallel h\parallel }_{{\mathcal{A}}_{G_{\Lambda }}}^2=\sum _{\lambda \in \Lambda }{\left|\widehat{\phi }\left(\lambda \right)\right|}^2.$$

(27)

Proof. 

We have that $G_{\Lambda }\left(x\right)={\sum }_{\lambda \in \Lambda }{e}^{i\lambda x}$, $x\in {\mathbb{R}}^k$ as a conditional exponential convex tempered distribution. We will prove that

$$\parallel \phi \ast G_{\Lambda }{\parallel }_{{\mathcal{A}}_{G_{\Lambda }}}^2=\sum _{\lambda \in \Lambda }{\left|\widehat{\phi }\left(\lambda \right)\right|}^2,$$

(28)

where $\phi \in S^k$ (the Schwartz space on ${\mathbb{R}}^k$), and $\widehat{\phi }$ is the standard Fourier transform. From (25), we have

$$\begin{array}{ccc}\hfill (\phi \ast G_{\Lambda })\left(x\right)& =& {\int }_{{\mathbb{R}}^k}\phi \left(y\right)G_{\Lambda }(x-y)dy\hfill \\ & =& {\int }_{{\mathbb{R}}^k}\phi \left(y\right)\sum _{\lambda \in \Lambda }{e}^{i\lambda (x-y)}dy\hfill \\ & =& \sum _{\lambda \in \Lambda }{\int }_{{\mathbb{R}}^k}\phi \left(y\right)e^{i\lambda (x-y)}dy\hfill \\ & =& \sum _{\lambda \in \Lambda }{\int }_{{\mathbb{R}}^k}\phi \left(y\right)e^{-i\lambda y}dy{e}^{i\lambda x}.\hfill \end{array}$$

Hence,

$$(\phi \ast G_{\Lambda })\left(x\right)=\sum _{\lambda \in \Lambda }\widehat{\phi }\left(\lambda \right)e^{i\lambda x}.$$

And so,

$$\begin{array}{ccc}\hfill \parallel \phi \ast G_{\Lambda }{\parallel }_{{\mathcal{A}}_{G_{\Lambda }}}^2& =& {⟨\phi \ast G_{\Lambda },\phi \ast G_{\Lambda }⟩}_{{\mathcal{A}}_{G_{\Lambda }}}\hfill \\ & =& {\int }_{{\mathbb{R}}^k}{\int }_{{\mathbb{R}}^k}{G}_{\Lambda }(x-y)\phi \left(x\right)\overline{\phi \left(y\right)}dxdy\hfill \\ & =& {\int }_{{\mathbb{R}}^k}{\int }_{{\mathbb{R}}^k}\sum _{\lambda \in \Lambda }{e}^{i\lambda (x-y)}\phi \left(x\right)\overline{\phi \left(y\right)}dxdy.\hfill \end{array}$$

Using Fubini’s theorem, we have

$$\begin{array}{ccc}\hfill \sum _{\lambda \in \Lambda }{\int }_{{\mathbb{R}}^k}{\int }_{{\mathbb{R}}^k}{e}^{i\lambda (x-y)}\phi \left(x\right)\overline{\phi \left(y\right)}dxdy& =& \sum _{\lambda \in \Lambda }\left[{\int }_{{\mathbb{R}}^k}{e}^{i\lambda x}\phi \left(x\right)dx{\int }_{{\mathbb{R}}^k}{e}^{-i\lambda y}\overline{\phi \left(y\right)}dy\right]\hfill \\ & =& \sum _{\lambda \in \Lambda }\left[{\int }_{{\mathbb{R}}^k}\overline{e^{-i\lambda x}\phi \left(x\right)}dx{\int }_{{\mathbb{R}}^k}{e}^{-i\lambda y}\phi \left(y\right)dy\right]\hfill \\ & =& \sum _{\lambda \in \Lambda }{\left|\widehat{\phi }\left(\lambda \right)\right|}^2.\hfill \end{array}$$

So,

$$\parallel \phi \ast G_{\Lambda }{\parallel }_{{\mathcal{A}}_{G_{\Lambda }}}^2=\sum _{\lambda \in \Lambda }{\left|\widehat{\phi }\left(\lambda \right)\right|}^2.$$

(29)

□

5. Generalized Fourier Transform on Infinite Spaces

In ([5], Chapter 4.3), the author introduces a generalized Fourier transform for infinite-dimensional spaces, specifically on the space $L_2(S^{*},P)$, where P denotes the measure derived from the Bochner–Minlos theorem. This approach is specialized to the case where the CECF is chosen as $G\left(s\right)={\parallel s\parallel }_{L_2}^2$. In this subsection, we extend the framework from [5] to consider a broader class of real-valued CECFs $G:S\to \mathbb{R}$, which are continuous under the Fréchet topology. This extension enables us to address a significantly wider range of Gaussian processes.

A key distinction between our method and that of [5] is that we provide a more direct definition of the generalized Fourier transform (refer to Equation (30)), which is closely linked to our approach for deriving Gaussian processes via the Bochner–Minlos theorem and the Gelfand triple. For each $H\in L_2(S^{*},P_{\lambda })$, we define a real-valued function $T_{\lambda }\left(H\right)$ on S according to the following rule:

$$T_{\lambda }\left(H\right)\left(s\right)=E_{P_{\lambda }}\left[Hexp\left[i{X}_s\right]\right]={\int }_{S^{*}}H\left(\omega \right)exp\left[i{X}_s\left(\omega \right)\right]d{P}_{\lambda }\left(\omega \right)={⟨H,exp\left[i{X}_s\right]⟩}_{L_2(S^{*},P_{\lambda })}.$$

(30)

In infinite dimensions, the transform $T_{\lambda }$ is the analog of the classical Fourier transform, where $X_s$ is the random variable on $S^{*}$, as $X_s\left(\omega \right)=⟨\omega ,s⟩=\omega \left(s\right)$, and $E_P$ is the expectation w.r.t. P.

Lemma 2.

For every $H\in L_2(S^{*},P_{\lambda })$, we have $T_{\lambda }\left(H\right)\in {\mathcal{A}}_{G_{\lambda }}$. Let ${\mathcal{A}}_0$ be a pre-Hilbert space generated by

$${\mathcal{A}}_0:=\left\{\sum _{j=1}^n{\alpha }_j{G}_{\lambda ,s_j},s_j\in S\right\},$$

(31)

where the RKHS ${\mathcal{A}}_{G_{\lambda }}$ is the completion of ${\mathcal{A}}_0$ with the inner product:

$${⟨\sum _{j=1}^n{\alpha }_j{G}_{\lambda ,s_j},\sum _{k=1}^n{\alpha }_k{G}_{\lambda ,s_k}⟩}_{{\mathcal{A}}_{G_{\lambda }}}=\sum _{j,k=1}^n{\alpha }_j{\alpha }_k{G}_{\lambda }(s_j-s_k)=G_{\lambda }\left(s\right).$$

(32)

Proof. 

Let $H\in L_2(S^{*},P_{\lambda })$; then, for every $n\in \mathbb{N}$, to prove $T_{\lambda }\left(H\right)\in {\mathcal{A}}_{G_{\lambda }}$, we have to prove that there exists $M_H>0$ such that:

$${∥\sum _{j=1}^n{\alpha }_j\left(T_{\lambda }\left(H\right)\right)\left(s_j\right)∥}^2\le M_H{∥\sum _{j=1}^n{\alpha }_j{G}_{\lambda ,s_j}∥}_{{\mathcal{A}}_{G_{\lambda }}}^2.$$

Let $H\in L_2(S^{*},P_{\lambda })$; then, for every $n\in \mathbb{N}$, we have

$$\begin{array}{ccc}\hfill {\left|\sum _{j=1}^n{\alpha }_j\left(T_{\lambda }\left(H\right)\right)\left(s_j\right)\right|}^2& =& E_{P_{\lambda }}{\left[H\sum _{j=1}^n{\alpha }_{j}exp\left[i{X}_{s_j}\right]\right]}^2\hfill \\ & =& {\left|{⟨H,\sum _{j=1}^n{\alpha }_{j}exp\left[i{X}_{s_j}\right]⟩}_{L_2(S^{*},P_{\lambda })}\right|}^2\hfill \\ & \le & M_H{∥\sum _{j=1}^n{\alpha }_{j}exp\left[i{X}_{s_j}\right]∥}_{L_2(S^{*},P_{\lambda })}^2\hfill \\ & =& M_H\sum _{j,k=1}^n{\alpha }_j{\alpha }_k{E}_{P_{\lambda }}\left[exp\left[i{X}_{s_j-s_k}\right]\right]\hfill \\ & =& M_H\sum _{j,k=1}^n{\alpha }_j{\alpha }_k{G}_{\lambda }(s_j-s_k).\hfill \end{array}$$

From [1], $G_{\lambda }\left(s\right)=E_{P_{\lambda }}\left[exp\left[i{X}_s\right]\right]$, let $M_H={\parallel H\parallel }_{L_2(S^{*},P_{\lambda })}^2<\infty$.

$$\begin{array}{ccc}\hfill M_H\sum _{j,k=1}^n{\alpha }_j{\alpha }_k{⟨G_{\lambda ,s_j},G_{\lambda ,s_k}⟩}_{{\mathcal{A}}_{G_{\lambda }}}& =& M_H{∥\sum _{j=1}^n{\alpha }_j{G}_{\lambda ,s_j}∥}_{{\mathcal{A}}_{G_{\lambda }}}^2.\hfill \end{array}$$

Thus, we conclude:

$${∥\sum _{j=1}^n{\alpha }_j\left(T_{\lambda }\left(H\right)\right)\left(s_j\right)∥}^2\le M_H{∥\sum _{j=1}^n{\alpha }_j{G}_{\lambda ,s_j}∥}_{{\mathcal{A}}_{G_{\lambda }}}^2.$$

(33)

So, $T_{\lambda }\left(H\right)\in {\mathcal{A}}_{G_{\lambda }}$. □

6. Concluding Remarks

This work explores the theoretical properties of CECFs and their implications in white noise analysis. By studying continuous CECFs, we establish their role in defining convolution semigroups and constructing RKHS. Additionally, we introduce a novel perspective on generalized Fourier transforms, connecting them to Gaussian processes through advanced functional analysis techniques. The presented framework extends beyond white noise analysis and offers new insights into structured function spaces. It provides a pathway for future research in abstract measure theory and its interaction with harmonic analysis. Potential applications may include the development of refined techniques for stochastic integration and spectral representations of convolution operators.