Offset Linear Canonical Stockwell Transform for Boehmians

Abstract

In this article, we construct a Boehmian space using the convolution theorem that contains the offset linear canonical Stockwell transforms (OLCST) of all square-integrable Boehmians. It is also proven that the extended OLCST on square-integrable Boehmians is consistent with the traditional OLCST. Furthermore, it is one-to-one, linear, and continuous with respect to $\Delta$-convergence as well as $\Delta$-convergence. Subsequently, we introduce a discrete variant of the OLCST. Ultimately, we broaden the application of the presented work by examining the OLCST within the domain of almost periodic functions.

1. Introduction

Offset Linear Canonical Transform (OLCT) [1,2,3,4] is a mathematical tool widely employed in signal processing, optics, and various scientific and engineering domains. It extends the capabilities of Linear Canonical Transform (LCT). Moshinsky and Quesne’s foundational work in quantum mechanics laid the formulation for LCT [5]. In contemporary signal processing research, LCT has emerged as a prominent area of study. This heightened interest has spurred substantial efforts to elucidate its mathematical foundations, leading to the establishment of numerous theorems pertaining to critical concepts such as sampling theorems [6,7,8], convolution theorems [9,10], and uncertainty principles [11]. The field of signal processing, in particular, has experienced a notable surge in investigations centered on LCT [12]. Originally, LCT was recognized for its ability to encompass fundamental transforms such as Fourier transform (FT) [13,14], fractional Fourier transform (FrFT) [15,16,17] and Fresnel transform [18]. These transformations have long been cornerstones in signal processing and optics, enabling the analysis and manipulation of signals and images in different domains.

Defined by a set of six parameters $(A,B,C,D,\tau ,\eta )$, OLCT, also known as special affine Fourier transform (SAFT) [19], represents the time-shifted $\tau$ and frequency-modulated $\eta$ version of the LCT [20], which has four parameters $(A,B,C,D)$. The additional parameters enhance the generality and flexibility of OLCT, making it applicable to a wide range of electrical and optical signal systems. OLCT has seen a rise in applications in various fields, such as optics and signal processing, in recent years [2,3,21]. In signal processing, FT represents signals widely in both temporal and spatial domains. While short-time Fourier transform (StFT) has addressed some limitations of Fourier transform, it struggles with signals featuring high frequencies for a small duration and low frequencies for a large duration because the analysis window width is fixed. Continuous wavelet transform (WT) [22,23] was introduced to overcome this limitation by offering greater adaptability. However, WT has its own drawbacks, such as a lack of phase information and poor directionality.

LCT faces challenges when analyzing signals with poorly concentrated energy in the frequency domain and non-stationary characteristics. Windowed linear canonical transform (WLCT) involves segmenting the signal before LCT spectral analysis to address these issues. However, the fixed window width restricts its practical application. Gupta et al. [24] introduced linear canonical wavelet transform (LCWT) as a solution, overcoming the limitations of LCT, WT, and WLCT.

To further address these challenges, Stockwell et al. [25] introduced Stockwell transform (ST) by merging advantages of the StFT and wavelet transform; the classical Stockwell transform provides accurate information about the local behavior of a signal in temporal–spatial analysis [26]. To analyse signals with multi-angle, multi-resolution, multi-scale, and temporal localization capabilities, researchers have proposed extensions of the Stockwell transform, such as fractional Stockwell transform (FrST) [27,28,29,30,31] and linear canonical Stockwell transform (LCST) [32,33,34].

In recent years, several expansions of Stockwell transform (ST) have emerged, integrating it with other relevant transforms such as FrFT, LCT, and Dunkl transforms, and others [27,32,34,35,36]. These extensions, such as Dunkl–Stockwell transform, fractional Stockwell transform (FrST), and linear canonical Stockwell transform (LCST) facilitate the analysis of signals with capabilities for multi-scale, multi-resolution, and temporal localization, etc. Consequently, they are adept at handling chirp-like signals, which are pervasive in various domains. Delving into the expansive realms of Stockwell transforms (STs) and recognizing the significant utility of the OLCT, there is a compelling drive to integrate their distinct advantages into an innovative transformation termed offset linear canonical Stockwell transform (OLCST). The main benefit of this suggested OLCST is that it inherits some interesting aspects from the classic ST and OLCT in addition to the good mathematical qualities of the former, especially when it comes to concurrently showing both temporal and OLCST domain information. The proposed transform offers increased degrees of freedom, improving its versatility in non-stationary signal processing applications, among other fields.

In 1981, Mikusinski [37] presented Boehmians as sequence quotients to generalize functions and distributions. Following the significant work by Mikusiński on the convergence of Boehmians [38], numerous studies [39,40,41,42,43,44,45] have been conducted by various researchers from different viewpoints. In this work, we apply OLCST to the space of square-integrable Boehmians. The space contains an example of a Boehmian that does not match to any distribution, as shown in [38]. Hence, defining OLCST on square-integrable Boehmians is a reasonable generalization of OLCST on $L^2\left(\mathbb{R}\right)$.

The structure of the paper is as given: Section 2 offers the necessary background, including OLCST. In Section 3, we extend OLCST to the space of square-integrable Boehmians and the definition of OLCST in this space serves as a proper generalization of OLCST on $L^2\left(\mathbb{R}\right).$ Section 4 discusses the discrete iteration of the new transform and derives its reconstruction formula. In Section 5, we expand our study to explore the application of the novel transform to almost periodic functions (APFs). In Section 6, we provide the potential applications, and finally, Section 7 provides the conclusions of our work.

2. Preliminaries

Let $L_p\left(\mathbb{R}\right)$ represent the space of all complex-valued Lebesgue measurable functions on $\mathbb{R}$ with

$${\Vert \mathrm{f}\Vert }_p={\left({\int }_{\mathbb{R}}{\left|\mathrm{f}\right(t\left)\right|}^{p}dt\right)}^{{\displaystyle \frac{1}{p}}}<\infty \mathrm{where}p=1,2.$$

We also represent the set of all complex-valued functions which are Lebesgue measurable by

$$\begin{array}{c}\hfill {\parallel \left|\Phi \right|\parallel }_2={\left({\int }_{\mathbb{R}}{\int }_{\mathbb{R}}{\left|\Phi (a,b)\right|}^{2}db{\displaystyle \frac{da}{\left|a\right|}}\right)}^{{\displaystyle \frac{1}{2}}}<\infty ,\end{array}$$

by ${\mathcal{L}}^2\left({\mathbb{R}}^2\right).$

Fix a mother affine wavelet $\varphi \in L^1\left(\mathbb{R}\right)\cup L^2\left(\mathbb{R}\right)$ such that ${\int }_{\mathbb{R}}\left({\displaystyle \frac{|\widehat{\varphi }{\left(t\right)|}^2}{\left|t\right|}}\right)dt<\infty ,$ where $\widehat{\varphi }\left(t\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{\mathbb{R}}\varphi \left(x\right)e^{-itx}dx,\forall t\in \mathbb{R},$ the Fourier transform of $\varphi$.

Definition 1. 

For any signal $\mathrm{f}\left(t\right)\in L^2\left(\mathbb{R}\right)$ and augmented matrix parameter $\mathbf{A}=\left(\begin{array}{ccc}A& B;& \tau \\ C& D;& \eta \end{array}\right)$ with $AD-BC=1,$ OLCT is given by

$$\begin{array}{c}\hfill {\mathbf{O}}_{\mathbf{A}}\left[\mathrm{f}\right(t)]\left(\omega \right)={\int }_{\mathbb{R}}\mathrm{f}(t){\mathcal{K}}_{\mathbf{A}}(t,\omega )dt,\end{array}$$

(1)

where ${\mathcal{K}}_{\mathbf{A}}(t,\omega )={\mathcal{K}}_B{e}^{{\displaystyle \frac{j}{2B}}(A{t}^2+2t(\tau -\omega )-2\omega (D\tau -B\eta )+D({\omega }^2+{\tau }^2))}$ with ${\mathcal{K}}_B={\displaystyle \frac{1}{\sqrt{j2\pi B}}}$. We only took into account the case in which B is not 0, as OLCT is only a chirp multiplication when $B=0$. The inversion formula corresponding to OLCT is given by

$$\begin{array}{c}\hfill {\mathbf{O}}_{{\mathbf{A}}^{-1}}\left[{\mathbf{O}}_{\mathbf{A}}\mathrm{f}\right]\left(t\right)=\mathrm{f}\left(t\right)=\mathcal{K}{\int }_{\mathbb{R}}{\mathbf{O}}_{\mathbf{A}}\left[\mathrm{f}\right]\left(\omega \right){\mathcal{K}}_{{\mathbf{A}}^{-1}}(\omega ,t)d\omega ,\end{array}$$

where ${\mathbf{A}}^{-1}=\left(\begin{array}{ccc}D& -B;& B\eta -D\tau \\ -C& A;& C\tau -A\eta \end{array}\right)$ and $\mathcal{K}=e^{{\displaystyle \frac{j}{2}}CD{\tau }^2-2AD\tau \eta +AB{\eta }^2}$. The Parseval’s formula is given by

$$\begin{array}{c}\hfill \langle \mathrm{f},\mathrm{g}\rangle =\langle {\mathbf{O}}_{\mathbf{A}}\left[\mathrm{f}\right]\left(\omega \right),{\mathbf{O}}_{\mathbf{A}}\left[\mathrm{g}\right]\left(\omega \right)\rangle .\end{array}$$

Definition 2. 

For $f\in L^2\left(\mathbb{R}\right),$ $\psi \in L^2\left(\mathbb{R}\right),$ and matrix parameter $\mathbf{A}=\left(\begin{array}{ccc}A& B;& \tau \\ C& D;& \eta \end{array}\right),$ the OLCST of f is defined as

$$\begin{array}{cc}\hfill {\mathbf{O}}_{\psi }^{\mathbf{A}}\left[f\right](a,b)& ={\int }_{\mathbb{R}}f\left(\mathrm{t}\right)\overline{{\psi }_{a,b}^{\mathbf{A}}\left(\mathrm{t}\right)}d\mathrm{t},\hfill \end{array}$$

(2)

where

$$\begin{array}{c}\hfill {\psi }_{a,b}^{\mathbf{A}}\left(\mathrm{t}\right)=a{e}^{{\displaystyle \frac{j}{2B}}(A{b}^2+2\tau b)}\psi \left(a(\mathrm{t}-b)\right)e^{ja\mathrm{t}-{\displaystyle \frac{j}{2B}}(A{\mathrm{t}}^2+2\tau \mathrm{t})}.\end{array}$$

(3)

The Parseval’s identity for OLCST is defined by

$$\begin{array}{c}\hfill {\int }_{\mathbb{R}}{\int }_{{\mathbb{R}}^{+}}{\mathbf{O}}_{\psi }^{\mathbf{A}}\left[\mathrm{f}\right](a,b)\overline{{\mathbf{O}}_{\psi }^{\mathbf{A}}\left[\mathrm{g}\right](a,b)}db{\displaystyle \frac{da}{a}}={\left|\mathcal{C}\right|}^2{\mathcal{C}}_{\Psi }\langle \mathrm{f},\mathrm{g}\rangle ,\end{array}$$

(4)

where ${\displaystyle {\mathcal{C}}_{\Psi }={\int }_{{\mathbb{R}}^{+}}{\left|\left({\mathbf{O}}_{\mathbf{A}}\Psi \right)\left(\frac{\omega }{a}\right)\right|}^2\frac{da}{a}.}$ The reconstruction formula is given by

$$\begin{array}{c}\hfill \mathrm{f}\left(t\right)={\displaystyle \frac{1}{{\left|\mathcal{C}\right|}^2{\mathcal{C}}_{\Psi }}}{\int }_{\mathbb{R}}{\int }_{{\mathbb{R}}^{+}}{\mathbf{O}}_{\psi }^{\mathbf{A}}\left[\mathrm{f}\right](a,b){\psi }_{a,b}^{\mathbf{A}}\left(t\right)db{\displaystyle \frac{da}{a}},a.e.\end{array}$$

(5)

3. Boehmian Space (B-S)

In this part, we revisit the construction of B-S and proceed to establish the B-S ${\mathcal{B}}_2\left({\mathcal{L}}^2\left({\mathbb{R}}^2\right),(\mathcal{D}\left(\mathbb{R}\right),\ast ),{⊛}_{\mathbf{A}},{\Delta }_0\right)$. This B-S encompasses the OLCSTs of square-integrable Boehmians.

Consider a semi-group (commutative) $(S,\times ),$ vector space $\Gamma$, and $★:\Gamma \times S\to \Gamma$, meeting the subsequent conditions:

(1)

$(\alpha +\beta )★\xi =(\alpha ★\xi )+(\beta ★\xi ),\forall \alpha ,\beta \in \Gamma$, and $\forall \xi \in S$.

(2)

$\left(c\alpha \right)★\xi =c(\alpha ★\xi ),\forall c\in \mathbb{C},\forall \alpha \in \Gamma$, and $\forall \xi \in S$.

(3)

$\alpha ★(\xi \times \zeta )=(\alpha ★\xi )★\zeta ,\forall \alpha \in \Gamma$, and $\forall \xi ,\zeta \in S$.

(4)

If $\xi \in S$ and ${\lim }_{n\to \infty }{\alpha }_n\to \alpha$ in $\Gamma$, then ${\lim }_{n\to \infty }{\alpha }_n★\xi \to \alpha ★\xi$ in $\Gamma .$

A set of all sequences $\left({\xi }_n\right)$ originating from $S,$ denoted by $\Delta$, that adhere to the following conditions:

$\left({\Delta }_1\right)$

if $\left({\xi }_n\right),\left({\zeta }_n\right)$ belongs to $\Delta$, then $\left({\xi }_n\times {\zeta }_n\right)\in \Delta$;

$\left({\Delta }_2\right)$

if ${\lim }_{n\to \infty }{\alpha }_n\to \alpha$ in $\Gamma$ and $\left({\xi }_n\right)\in \Delta$, then ${\lim }_{n\to \infty }{\alpha }_n★{\xi }_n\to \alpha$ in $\Gamma$.

All Boehmians are expressed in the form $\left[\left({\alpha }_n\right)/\left({\xi }_n\right)\right]$, where ${\alpha }_n\in \Gamma$ and $\left({\xi }_n\right)\in \Delta \forall n\in \mathbb{N}$, satisfying the following condition:

$$\begin{array}{c}\hfill {\alpha }_n★{\xi }_m={\alpha }_m★{\xi }_n\mathrm{for}\mathrm{all}m,n\in \mathbb{N}.\end{array}$$

The collective set of all Boehmians is denoted as $\mathcal{B}=\mathcal{B}(\Gamma ,(S,\times ),★,\Delta )$. Two Boehmians, $\left[\left({\alpha }_n\right)/\left({\xi }_n\right)\right]$ and $\left[\left({\beta }_n\right)/\left({\zeta }_n\right)\right]$ in $\mathcal{B}$, are considered equal if

$$\begin{array}{c}\hfill {\alpha }_n★{\zeta }_m={\beta }_m★{\xi }_n\forall m,n\in \mathbb{N}.\end{array}$$

The space $\Gamma$ is mapped to a subset of $\mathcal{B}$ via the mapping $\alpha \mapsto \left[\left(\alpha ★{\xi }_n\right)/\left({\xi }_n\right)\right]$, where $\left({\xi }_n\right)\in \Delta$ is arbitrary. In the sense of $\mathcal{B}$, properties of operation ★ are defined as follows:

• Addition: $\left[\left({\alpha }_n\right)/\left({\xi }_n\right)\right]+\left[\left({\beta }_n\right)/\left({\zeta }_n\right)\right]=\left[\left({\alpha }_n★{\zeta }_n+{\beta }_n★{\xi }_n\right)/\left({\xi }_n\times {\zeta }_n\right)\right];$
• Scalar Multiplication: $c\left[\left({\alpha }_n\right)/\left({\xi }_n\right)\right]=\left[\left(c{\alpha }_n\right)/\left({\xi }_n\right)\right];$
• $\left[\left({\alpha }_n\right)/\left({\xi }_n\right)\right]★\xi =\left[\left({\alpha }_n★\xi \right)/\left({\xi }_n\right)\right].$

Two convergence notions for B-Ss are described as follows.

Definition 3. 

A Boehmian sequence $\left(X_n\right)$ is δ-converged to X if $\exists \left({\xi }_k\right)\in \Delta$, so that $X_n★{\xi }_k\in \Gamma \forall n,k$ and $X★{\xi }_k\in \Gamma \forall k$, and for each k and n that approaches ∞, $X_n★{\xi }_k\to X★{\xi }_k$ in Γ.

Definition 4. 

A Boehmian sequence $\left(X_n\right)$ is Δ-converged to X if $\exists \left({\xi }_n\right)\in \Delta$, so that $\left(X_n-X\right)★{\xi }_n\in \Gamma \forall n$, and $\left(X_n-X\right)★{\xi }_n\to 0$ as n approaches ∞ in Γ.

The construction of ${\mathcal{L}}^p$-Boehmians, as detailed in [46] for $p>1$, focuses on recalling the definition of Boehmians that are square-integrable. The space ${\mathcal{BL}}^2\left(\mathbb{R}\right)$ denotes the collection of square integrable Boehmians and can be represented as $\mathcal{B}(L^2\left(\mathbb{R}\right),(\mathcal{D}\left(\mathbb{R}\right),{\ast }_{\mathbf{A}}),{\ast }_{\mathbf{A}},{\Delta }_0)$. This representation arises by considering $\Gamma$ as $L^2\left(\mathbb{R}\right)$. The semi-group (commutative) $(S,\times )$ is specified as $(\mathcal{D}\left(\mathbb{R}\right),{\ast }_{\mathbf{A}})$, where $\mathcal{D}\left(\mathbb{R}\right)$ denotes the Schwartz space on $\mathbb{R}$. In this context, ${\ast }_{\mathbf{A}}$ denotes the convolution operation functions, and the space is defined by

$$\begin{array}{c}\hfill \left(\mathrm{f}{\ast }_{\mathbf{A}}\mu \right)\left(t\right)=e^{{\displaystyle \frac{-i}{2B}}(A{t}^2+2\tau t)}(\tilde{\mathrm{f}}\ast \tilde{\mu })\left(t\right),\end{array}$$

where $\tilde{f}\left(t\right)=e^{{\displaystyle \frac{i}{2B}}(A{t}^2+2\tau t)}$ and ∗ is classical convolution.

Therefore,

$$\left(\mathrm{f}{\ast }_{\mathbf{A}}\mu \right)\left(t\right)=e^{{\displaystyle \frac{-i}{2B}}(A{t}^2+2\tau t)}{\int }_{-\infty }^{\infty }\mathrm{f}(t-s)e^{{\displaystyle \frac{i}{2B}}(A{(t-s)}^2+2\tau (t-s))}\mu \left(s\right)e^{{\displaystyle \frac{i}{2B}}(A{s}^2+2\tau s)}ds\forall x\in \mathbb{R}.$$

Here, ★ denotes the identical convolution operation ${\ast }_{\mathbf{A}}$, and $\Delta$ represents the collection denoted as ${\Delta }_0$, encompassing all sequences $\left({\mu }_n\right)$ originating from $\mathcal{D}\left(\mathbb{R}\right)$ that adhere to the following properties:

(1)

${\int }_{-\infty }^{\infty }{\mu }_n\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}=1,\forall n\in \mathbb{N}$;

(2)

${\int }_{-\infty }^{\infty }\left|{\mu }_n\left(\mathrm{t}\right)\right|\mathrm{d}\mathrm{t}\le \mathrm{M},\forall n\in \mathbb{N}$ for some $\mathrm{M}>0$;

(3)

For the given $ϵ>0$, $\exists m\in \mathbb{N}$ such that supp ${\mu }_n\subset (-ϵ,ϵ),\forall n\ge m$, where supp ${\mu }_n$ is the support of ${\mu }_n$.

Subsequently, we formulate the B-S denoted as ${\mathcal{B}}_{{\mathcal{L}}^2\left({\mathbb{R}}^2\right)}=\mathcal{B}\left({\mathcal{L}}^2\left({\mathbb{R}}^2\right),(\mathcal{D}\left(\mathbb{R}\right),{\ast }_{\mathbf{A}}),{⊛}_{\mathbf{A}},{\Delta }_0\right)$ by introducing the operation ${⊛}_{\mathbf{A}}$ and establishing its essential properties through rigorous proof.

Definition 5. 

For $\Phi \in {\mathcal{L}}^2\left({\mathbb{R}}^2\right)$ and $\mu \in \mathcal{D}\left(\mathbb{R}\right)$, define

$$\begin{array}{c}\hfill \left(\Phi {⊛}_{\mathbf{A}}\mu \right)(v,u)={\int }_{-\infty }^{\infty }\Phi (v-s,u)\mu \left(s\right){\mathrm{e}}^{-\mathrm{i}su+{\displaystyle \frac{i}{2B}}(A{s}^2+2\tau s)}ds\forall (v,u)\in {\mathbb{R}}^2.\end{array}$$

Lemma 1. 

If $\Phi \in {\mathcal{L}}^2\left({\mathbb{R}}^2\right)$ and $\mu \in \mathcal{D}\left(\mathbb{R}\right),$ then $\parallel \Phi {⊛}_{\mathbf{A}}{\mu \parallel }_2\le {\parallel \Phi \parallel }_2{\parallel \mu \parallel }_1$, and hence $\Phi {⊛}_{\mathbf{A}}\mu \in L^2\left({\mathbb{R}}^2\right)$.

Proof. 

For ${\left|\mu \right|}_1=0$, the lemma follows trivially. Therefore, we proceed under the assumption that ${\left|\mu \right|}_1\ne 0$. By applying Jensen’s property and Fubini’s theorem, we obtain

$$\begin{array}{cc}\hfill \parallel |\Phi {⊛}_{\mathbf{A}}\mu {|\parallel }_2^2& \le {\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\left({\int }_{-\infty }^{\infty }\left|\Phi (v-s,u)\mu \left(s\right)\right|\mathrm{d}s\right)}^2\mathrm{d}v{\displaystyle \frac{\mathrm{d}u}{\left|u\right|}}\hfill \\ \hfill & \le {\parallel \mu \parallel }_1^2{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\left({\int }_{-\infty }^{\infty }{\left|\Phi (v-s,u)\right|}^2\left|\mu \left(s\right)\right|{\displaystyle \frac{\mathrm{d}s}{{\parallel \mu \parallel }_1}}\right)\mathrm{d}v{\displaystyle \frac{\mathrm{d}u}{\left|u\right|}},\hfill \end{array}$$

Given that $\left|\mu \left(s\right)\right|{\displaystyle \frac{\mathrm{d}s}{{\left|\mu \right|}_1}}$ forms a probability measure on $\mathbb{R}$, Jensen’s inequality can be applied:

$$\begin{array}{cc}\hfill \parallel \Phi {⊛}_{\mathbf{A}}{\mu \parallel }_2^2& \le {\parallel \mu \parallel }_1{\int }_{-\infty }^{\infty }\left|\mu \left(s\right)\right|{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\left|\Phi (v-s,u)\right|}^2\mathrm{d}v{\displaystyle \frac{\mathrm{d}u}{\left|u\right|}}\mathrm{d}s\hfill \\ \hfill & ={\parallel \mu \parallel }_1^2{\parallel \left|\Phi \right|\parallel }_2^2.\hfill \end{array}$$

Hence, it is proven. □

The subsequent lemma directly results from Lemma 1.

Lemma 2. 

If ${\Phi }_n\to \Phi$ in ${\mathcal{L}}^2\left({\mathbb{R}}^2\right)$ as $n\to \infty$ and $\mu \in \mathcal{D}\left(\mathbb{R}\right)$, then ${\Phi }_n{⊛}_{\mathbf{A}}\mu \to \Phi {⊛}_{\mathbf{A}}\mu$ in ${\mathcal{L}}^2\left({\mathbb{R}}^2\right)$ as $n\to \infty$.

Lemma 3. 

If $\Phi ,{\Phi }_1,{\Phi }_2\in {\mathcal{L}}^2\left({\mathbb{R}}^2\right),\mu ,{\mu }_1,{\mu }_2\in \mathcal{D}\left(\mathbb{R}\right)$ and $c\in \mathbb{C}$, then

(1)

$\left({\Phi }_1+{\Phi }_2\right){⊛}_{\mathbf{A}}\mu ={\Phi }_1{⊛}_{\mathbf{A}}\mu +{\Phi }_2{⊛}_{\mathbf{A}}\mu$.

(2)

$\left(c\Phi \right){⊛}_{\mathbf{A}}\mu =c\left(\Phi {⊛}_{\mathbf{A}}\mu \right)$.

(3)

$\Phi {⊛}_{\mathbf{A}}\left({\mu }_1{\ast }_{\mathbf{A}}{\mu }_2\right)=\left(\Phi {⊛}_{\mathbf{A}}{\mu }_1\right){⊛}_{\mathbf{A}}{\mu }_2$.

Proof. 

Given that the first two identities follow directly, we focus our attention solely on demonstrating the validity of the third identity.

$$\begin{array}{cc}\hfill & \Phi {⊛}_{\mathbf{A}}\left({\mu }_1{\ast }_{\mathbf{A}}{\mu }_2\right)(v,u)\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }\Phi (v-s,u)\left({\mu }_1{\ast }_{\mathbf{A}}{\mu }_2\right)\left(s\right){\mathrm{e}}^{-isu+{\displaystyle \frac{i}{2B}}(A{s}^2+2\tau s)}ds\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }\Phi (v-s,u)e^{{\displaystyle \frac{-i}{2B}}(A{s}^2+2\tau s)}{\int }_{-\infty }^{\infty }{\mu }_1(s-t)e^{{\displaystyle \frac{i}{2B}}(A{(s-t)}^2+2\tau (s-t))}{\mu }_2\left(t\right)e^{{\displaystyle \frac{i}{2B}}(A{t}^2+2\tau t)}dt{\mathrm{e}}^{-isu+{\displaystyle \frac{i}{2B}}(A{s}^2+2\tau s)}ds\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\Phi (v-s,u){\mu }_1(s-T)e^{{\displaystyle \frac{i}{2B}}(A{(s-t)}^2+2\tau (s-t))}{\mathrm{e}}^{-isu}ds{\mu }_2\left(t\right)e^{{\displaystyle \frac{i}{2B}}(A{t}^2+2\tau t)}dt,\hfill \end{array}$$

using the change of variable $z=s-t$

$$\begin{array}{cc}\hfill \Phi {⊛}_{\mathbf{A}}\left({\mu }_1{\ast }_{\mathbf{A}}{\mu }_2\right)(v,u)& ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\Phi (v-(z+t),u){\mu }_1\left(z\right)e^{{\displaystyle \frac{i}{2B}}(A{z}^2+2\tau z)}{\mathrm{e}}^{-\mathrm{i}(z+t)u}\mathrm{d}z{\mu }_2\left(t\right)e^{{\displaystyle \frac{i}{2B}}(A{t}^2+2\tau t)}t\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\Phi ((v-t)-z),u){\mu }_1\left(z\right){\mathrm{e}}^{-\mathrm{i}zu+{\displaystyle \frac{i}{2B}}(A{z}^2+2\tau z)}dz{\mu }_2\left(t\right){\mathrm{e}}^{-\mathrm{i}tu+{\displaystyle \frac{i}{2B}}(A{t}^2+2\tau t)}dt\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }\left(\Phi {⊛}_{\mathbf{A}}{\mu }_1\right)(v-t,u){\mu }_2\left(t\right){\mathrm{e}}^{-\mathrm{i}tu+{\displaystyle \frac{i}{2B}}(A{t}^2+2\tau t)}dt\hfill \\ \hfill & =\left(\left(\Phi {⊛}_{\mathbf{A}}{\mu }_1\right){⊛}_{\mathbf{A}}{\mu }_2\right)(v,u)\mathrm{for}\mathrm{almost}\mathrm{all}u\in \mathbb{R}.\hfill \end{array}$$

Hence, $\Phi {⊛}_{\mathbf{A}}\left({\mu }_1{\ast }_{\mathbf{A}}{\mu }_2\right)=\left(\Phi {⊛}_{\mathbf{A}}{\mu }_1\right){⊛}_{\mathbf{A}}{\mu }_2$ in ${\mathcal{L}}^2\left({\mathbb{R}}^2\right)$. □

Lemma 4. 

If ${\Phi }_n\to \Phi$ in ${\mathcal{L}}^2\left({\mathbb{R}}^2\right)$ as $n\to \infty$ and $\left({\mu }_n\right)\in {\Delta }_0$, then ${\Phi }_n{⊛}_{\mathbf{A}}{\mu }_n\to \Phi$ in ${\mathcal{L}}^2\left({\mathbb{R}}^2\right)$ as $n\to \infty$.

Proof. 

Applying Lemma 3, we obtain

$$\begin{array}{c}\hfill {∥|{\Phi }_n{⊛}_{\mathbf{A}}{\mu }_n-\Phi |∥}_2\le {∥|\left({\Phi }_n-\Phi \right){⊛}_{\mathbf{A}}{\mu }_n|∥}_2+{∥|\Phi {⊛}_{\mathbf{A}}{\mu }_n-\Phi |∥}_2.\end{array}$$

(6)

Using Lemma 1 and property (2) of $\left({\mu }_n\right)\in {\Delta }_0$ on the right-hand side of (6), we observe that the initial term is bounded by $\mathrm{M}\parallel \left|{\Phi }_n-\Phi \right|{\parallel }_2$, which diminishes as $n\to \infty$. Our subsequent objective is to demonstrate the convergence of the second term towards zero.

Let $ϵ>0.$ We opt for $\Psi \in C_c(\mathbb{R}\times (\mathbb{R}\backslash 0))$, such that $\left|\right|\Phi -{\Psi \left|\right|}_2<ϵ$, leveraging the density property of the space $C_c(\mathbb{R}\times (\mathbb{R}\backslash 0))$, which encompasses all continuous functions with compact support on ${\mathbb{R}}^2$, in ${\mathcal{L}}^2(\mathbb{R}\times (\mathbb{R}\backslash 0))={\mathcal{L}}^2\left({\mathbb{R}}^2\right)$. For any given $(v,u)\in {\mathbb{R}}^2$, if we define $g\left(t\right)=\Psi (v-t,u){\mathrm{e}}^{-itu+{\displaystyle \frac{i}{2B}}(A{t}^2+2\tau t)}$ for all $t\in {\mathbb{R}}^2$, then $\mathrm{g}\in C_c\left(\mathbb{R}\right)$, signifying that g is uniformly continuous on $\mathbb{R}$. Consequently, there exists $\delta >0$ such that $\left|g\right(u)-g(v\left)\right|<ϵ$ whenever $v,u\in \mathbb{R}$ and $|v-u|\le \delta$. As supp ${\mu }_n\to 0$ is as $n\to \infty$, we can find $N\in \mathbb{N}$ such that supp ${\mu }_n$ is encompassed within the closed ball with the center at the origin and radius $\delta$ in ${\mathbb{R}}^2$, for all $n\ge N$. Now, we have the following:

• If supp $\Psi \subset [p,q]\times \mathrm{K}$ for some $-\infty <p<q<+\infty$ and a compact subset $\mathrm{K}$ of $\mathbb{R}\backslash \left\{0\right\}$, then $\Psi (v-s,u)=0$, for every $(v,u)\notin [p-\delta ,q+\delta ]\times \mathrm{K}$ and for every $y\in [-\delta ,\delta ]$.
• From property (2) of $\left({\mu }_n\right)\in {\Delta }_0$, we have ${∥{\mu }_n∥}_1\le \mathrm{M},\forall n\in \mathbb{N}$, for $\mathrm{M}>0$.

Therefore, by employing Jensen’s property, for $n\ge N$, we obtain

$$\begin{array}{cc}\hfill {∥|\Psi {⊛}_{\mathbf{A}}{\mu }_n-\Psi |∥}_2^2& ={\int }_K{\int }_{p-\delta }^{q+\delta }{\left({\int }_{-\infty }^{\infty }\left[\Psi (v-s,u){\mathrm{e}}^{-isu+{\displaystyle \frac{i}{2B}}(A{s}^2+2\tau s)}-\Psi (v,u)\right]{\mu }_n\left(s\right)\mathrm{d}s\right)}^{2}dv{\displaystyle \frac{\mathrm{d}u}{\left|u\right|}}\hfill \\ \hfill & ={\int }_K{\int }_{p-\delta }^{q+\delta }{\left({\int }_{-\infty }^{\infty }[g\left(s\right)-g\left(0\right)]{\mu }_n\left(s\right)\mathrm{d}s\right)}^{2}dv{\displaystyle \frac{\mathrm{d}u}{\left|u\right|}}\hfill \\ \hfill & \le {∥{\mu }_n∥}_1{\int }_K{\int }_{p-\delta }^{q+\delta }{\int }_{-\delta }^{\delta }{|g\left(s\right)-g\left(0\right)|}^2\left|{\mu }_n\left(s\right)\right|\mathrm{d}s\mathrm{d}v{\displaystyle \frac{\mathrm{d}u}{\left|u\right|}}\hfill \\ \hfill & \le {\mathrm{M}}^2{ϵ}^2{\int }_K{\int }_{p-\delta }^{q+\delta }\mathrm{d}b{\displaystyle \frac{\mathrm{d}a}{\left|a\right|}}={\displaystyle \frac{1}{\theta }}{\mathrm{M}}^2\mathrm{m}\left(\mathrm{K}\right)(q-p+2\delta ){ϵ}^2,\hfill \end{array}$$

Here, $\theta <0={\inf }_{a\in \mathrm{K}}\left|a\right|$ and $\mathrm{m}\left(\mathrm{K}\right)$ represent the Lebesgue measure of $\mathrm{K}$. Utilizing Lemma 1 and the calculated approximation, we obtain

$$\begin{array}{cc}\hfill {∥|{\Phi }_n{⊛}_{\mathbf{A}}{\mu }_n-\Phi |∥}_2& \le {∥|\left(\Phi -\Psi \right){⊛}_{\mathbf{A}}{\mu }_n|∥}_2+{∥|\Psi {⊛}_{\mathbf{A}}{\mu }_n-\Psi |∥}_2+{∥|\Psi -\Phi |∥}_2\hfill \\ \hfill & <\left([1+\sqrt{{\displaystyle \frac{1}{\theta }}m\left(K\right)(q-p+2\delta )}]M+1\right)ϵ.\hfill \end{array}$$

Thus, the lemma is concluded. □

Theorem 1. 

If a function $f\in L^2\left(\mathbb{R}\right)$ and $\mu \in \mathcal{D}\left(\mathbb{R}\right)$, then ${\mathbf{O}}_{\psi }^{\mathbf{A}}\left(f{\ast }_{\mathbf{A}}\mu \right)={\mathbf{O}}_{\psi }^{\mathbf{A}}f{⊛}_{\mathbf{A}}\mu$.

Proof. 

Let $(v,u)\in {\mathbb{R}}^2$ be arbitrary:

$$\begin{array}{cc}\hfill {\mathbf{O}}_{\psi }^{\mathbf{A}}\left(\mathrm{f}{\ast }_{\mathbf{A}}\mu \right)(v,u)& =a{e}^{{\displaystyle \frac{-i}{2B}}(A{b}^2+2\tau b)}{\int }_{-\infty }^{\infty }{\mathrm{e}}^{-\mathrm{i}tu+{\displaystyle \frac{i}{2B}}(A{t}^2+2\tau t)}\left(\mathrm{f}{\ast }_{\mathbf{A}}\mu \right)\left(t\right)\overline{\psi \left(u\right(t-v\left)\right)}\mathrm{d}t\hfill \\ \hfill & =a{e}^{{\displaystyle \frac{-i}{2B}}(A{b}^2+2tb)}{\int }_{-\infty }^{\infty }{\mathrm{e}}^{-itu}{\int }_{-\infty }^{\infty }\mathrm{f}(t-s)e^{{\displaystyle \frac{i}{2B}}(A{(t-s)}^2+2\tau (t-s))}\mu \left(s\right)e^{{\displaystyle \frac{i}{2B}}(A{s}^2+2\tau s)}ds\overline{\psi \left(u\right(t-v\left)\right)}\mathrm{d}t\hfill \\ \hfill & =a{e}^{{\displaystyle \frac{-i}{2B}}(A{b}^2+2\tau b)}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\mathrm{e}}^{-\mathrm{i}tu}\mathrm{f}(t-s)e^{{\displaystyle \frac{i}{2B}}(A{(t-s)}^2+2\tau (t-s))}\overline{\psi \left(u\right(t-v\left)\right)}dt\mu \left(s\right)e^{{\displaystyle \frac{i}{2B}}(A{s}^2+2\tau s)}ds\hfill \\ \hfill & =a{e}^{{\displaystyle \frac{-i}{2B}}(A{b}^2+2\tau b)}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\mathrm{e}}^{-\mathrm{i}(z+s)u}\mathrm{f}\left(z\right)e^{{\displaystyle \frac{i}{2B}}(A{z}^2+2\tau z)}\overline{\psi \left(u\right(z-(v-s)\left)\right)}dz\mu \left(s\right)e^{{\displaystyle \frac{i}{2B}}(A{s}^2+2\tau s)}\mathrm{d}s\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }({\mathbf{O}}_{\psi }^{\mathbf{A}}\mathrm{f})(v-s,u)\mu \left(s\right){\mathrm{e}}^{-\mathrm{i}su}{e}^{{\displaystyle \frac{i}{2B}}(A{s}^2+2\tau s)}ds\hfill \\ \hfill & =\left({\mathbf{O}}_{\psi }^{\mathbf{A}}\mathrm{f}{⊛}_{\mathbf{A}}\mu \right)(v,u).\hfill \end{array}$$

Hence, the theorem follows. □

Definition 6. 

We define the extended Stockwell transform ${\mathfrak{O}}_{\psi }^{\mathbf{A}}:{\mathcal{B}}_{L^2\left(\mathbb{R}\right)}\to {\mathcal{B}}_{{\mathcal{L}}^2\left({\mathbb{R}}^2\right)}$ by

$$\begin{array}{c}\hfill {\mathfrak{O}}_{\psi }^{\mathbf{A}}\left[\left({\mathrm{f}}_n\right)/\left({\mu }_n\right)\right]=\left[\left({\mathbf{O}}_{\psi }^{\mathbf{A}}{\mathrm{f}}_n\right)/\left({\mu }_n\right)\right],\forall \left[\left({\mathrm{f}}_n\right)/\left({\mu }_n\right)\right]\in {\mathcal{B}}_{L^2\left(\mathbb{R}\right)}.\end{array}$$

The extension of OLCST onto $\mathcal{B}{L}^2\left(\mathbb{R}\right)$ is clearly well defined. To elaborate, suppose $X=\left[\left({\mathrm{f}}_n\right)/\left({\mu }_n\right)\right]\in$$\mathcal{B}{L}^2\left(\mathbb{R}\right)$, and then

$${\mathrm{f}}_n{\ast }_{\mathbf{A}}{\mu }_m={\mathrm{f}}_m{\ast }_{\mathbf{A}}{\mu }_n\forall m,n\in \mathbb{N}.$$

Applying OLCST on both sides, we obtain

$${\mathbf{O}}_{\psi }^{\mathbf{A}}{\mathrm{f}}_n{⊛}_{\mathbf{A}}{\mu }_m={\mathbf{O}}_{\psi }^{\mathbf{A}}{\mathrm{f}}_m{⊛}_{\mathbf{A}}{\mu }_n\forall m,n\in \mathbb{N},$$

and hence $\left[\left({\mathbf{O}}_{\psi }^{\mathbf{A}}{\mathrm{f}}_n\right)/\left({\mu }_n\right)\right]\in {\mathcal{B}}_{{\mathcal{L}}^2\left({\mathbb{R}}^2\right)}$. Furthermore, if $\left[\left({\mathrm{f}}_n\right)/\left({\mu }_n\right)\right]=\left[\left(g_n\right)/\left({\lambda }_n\right)\right]\in {\mathcal{B}}_{L^2\left(\mathbb{R}\right)}$, then we have

$$\begin{array}{c}\hfill {\mathrm{f}}_n{\ast }_{\mathbf{A}}{\lambda }_m=g_m{\ast }_{\mathbf{A}}{\mu }_n\forall n,m\in \mathbb{N}.\end{array}$$

Using OLCST on both sides once more, we find that

$$\begin{array}{c}\hfill {\mathbf{O}}_{\psi }^{\mathbf{A}}{\mathrm{f}}_n{⊛}_{\mathbf{A}}{\lambda }_m={\mathbf{O}}_{\psi }^{\mathbf{A}}{g}_m{⊛}_{\mathbf{A}}{\mu }_n\forall m,n\in \mathbb{N}.\end{array}$$

This demonstrates that ${\mathfrak{O}}_{\psi }^{\mathbf{A}}X$ remains unaffected by the choice of representative for X.

Theorem 2. 

The extended OLCST ${\mathfrak{O}}_{\psi }^{\mathbf{A}}:{\mathcal{B}}_{L^2\left(\mathbb{R}\right)}\to {\mathcal{B}}_{{\mathcal{L}}^2\left({\mathbb{R}}^2\right)}$ is consistent with ${\mathbf{O}}_{\psi }^{\mathbf{A}}:$$L^2\left(\mathbb{R}\right)\to {\mathcal{L}}^2\left({\mathbb{R}}^2\right)$.

Proof. 

Consider any arbitrary function $\mathrm{f}\in L^2\left(\mathbb{R}\right)$. The Boehmian representing f in ${\mathcal{B}}_{L^2\left(\mathbb{R}\right)}$ is expressed as $[\left(\mathrm{f}{\ast }_{\mathbf{A}}{\mu }_n\right)/\left({\mu }_n\right)]$, where $\left({\mu }_n\right)\in {\Delta }_0$ is chosen arbitrarily. Consequently,

$${\mathfrak{O}}_{\psi }^{\mathbf{A}}\left[\left(\mathrm{f}\ast {\mu }_n\right)/\left({\mu }_n\right)\right]=\left[\left({\mathbf{O}}_{\psi }^{\mathbf{A}}\left(\mathrm{f}\ast {\mu }_n\right)\right)/\left({\mu }_n\right)\right]=\left[\left({\mathbf{O}}_{\psi }^{\mathbf{A}}\mathrm{f}{⊛}_{\mathbf{A}}{\mu }_n\right)/\left({\mu }_n\right)\right],$$

which is the Boehmian in ${\mathcal{B}}_{{\mathcal{L}}^2\left({\mathbb{R}}^2\right)}$, which represents ${\mathbf{O}}_{\psi }^{\mathbf{A}}\mathrm{f}$.      □

Theorem 3. 

The extended OLCST ${\mathfrak{O}}_{\psi }^{\mathbf{A}}:{\mathcal{B}}_{L^2\left(\mathbb{R}\right)}\to {\mathcal{B}}_{{\mathcal{L}}^2\left({\mathbb{R}}^2\right)}$ is linear.

Proof. 

The demonstration of the above theorem is obvious. □

Theorem 4. 

The extended OLCST ${\mathfrak{O}}_{\psi }^{\mathbf{A}}:{\mathcal{B}}_{L^2\left(\mathbb{R}\right)}\to {\mathcal{B}}_{{\mathcal{L}}^2\left({\mathbb{R}}^2\right)}$ is one–one.

Proof. 

Let $X=\left[\left({\mathrm{f}}_n\right)/\left({\mu }_n\right)\right],Y=\left[\left(g_n\right)/\left({\lambda }_n\right)\right]\in {\mathcal{B}}_{L^2\left(\mathbb{R}\right)}$ be such that ${\mathfrak{O}}_{\psi }^{\mathbf{A}}X={\mathfrak{O}}_{\psi }^{\mathbf{A}}Y$, that is, $\left[\left({\mathbf{O}}_{\psi }^{\mathbf{A}}{\mathrm{f}}_n\right)/\left({\mu }_n\right)\right]=\left[\left({\mathbf{O}}_{\psi }^{\mathbf{A}}{g}_n\right)/\left({\lambda }_n\right)\right]$. Therefore, it follows that

$${\mathbf{O}}_{\psi }^{\mathbf{A}}{\mathrm{f}}_n{⊛}_{\mathbf{A}}{\lambda }_m={\mathbf{O}}_{\psi }^{\mathbf{A}}{g}_m{⊛}_{\mathbf{A}}{\mu }_n\forall m,n\in \mathbb{N},$$

and therefore, according to Theorem 1, we obtain

$${\mathbf{O}}_{\psi }^{\mathbf{A}}\left({\mathrm{f}}_n\ast {\lambda }_m\right)={\mathbf{O}}_{\psi }^{\mathbf{A}}\left(g_m\ast {\mu }_n\right).$$

Since ${\mathbf{O}}_{\psi }^{\mathbf{A}}$ is one–one, we obtain

$$f_n{\ast }_{\mathbf{A}}{\lambda }_m=g_m{\ast }_{\mathbf{A}}{\mu }_n.$$

Thus, $X=\left[\left({\mathrm{f}}_n\right)/\left({\mu }_n\right)\right]=\left[\left(g_n\right)/\left({\lambda }_n\right)\right]=Y$. □

Theorem 5. 

The range of the extended OLCST ${\mathfrak{O}}_{\psi }^{\mathbf{A}}:{\mathcal{B}}_{\mathbb{R}}^2\to {\mathcal{B}}_{{\mathcal{L}}^2\left({\mathbb{R}}^2\right)}$ is

$$\left\{X\in {\mathcal{B}}_{{\mathcal{L}}^2\left({\mathbb{R}}^2\right)}:X\mathit{has}a\mathit{representation}\left[\left({\Phi }_n\right)/\left({\mu }_n\right)\right]\mathit{with}{\Phi }_n\in {\mathbf{O}}_{\psi }^{\mathbf{A}}\left(L^2\left(\mathbb{R}\right)\right)\forall n\in \mathbb{N}\right\}.$$

Proof. 

If $X\in {\mathfrak{O}}_{\psi }^{\mathbf{A}}\left({\mathcal{B}}_{\mathbb{R}}^2\right)$, then $\exists \left[\left({\mathrm{f}}_n\right)/\left({\mu }_n\right)\right]\in {\mathcal{B}}_{\mathbb{R}}^2$ such that ${\mathfrak{O}}_{\psi }^{\mathbf{A}}\left[\left({\mathrm{f}}_n\right)/\left({\mu }_n\right)\right]=X$. It is evident that $\left[\left({\mathbf{O}}_{\psi }^{\mathbf{A}}{\mathrm{f}}_n\right)/\left({\mu }_n\right)\right]$ works as an appropriate representation for X. Alternatively, consider $\left[\left({\Phi }_n\right)/\left({\mu }_n\right)\right]\in {\mathcal{B}}_{{\mathcal{L}}^2\left({\mathbb{R}}^2\right)}$ such that ${\Phi }_n\in {\mathbf{O}}_{\psi }^{\mathbf{A}}\left(L^2\left(\mathbb{R}\right)\right)$ for all $n\in \mathbb{N}$ and $\exists {\mathrm{f}}_n\in L^2\left(\mathbb{R}\right)$ such that ${\mathbf{O}}_{\psi }^{\mathbf{A}}{\mathrm{f}}_n={\Phi }_n$. Our claim is that $\left[\left({\mathrm{f}}_n\right)\left({\mu }_n\right)\right]\in {\mathcal{B}}_{\mathbb{R}}^2$. From $\left[\left({\Phi }_n\right)/\left({\mu }_n\right)\right]\in {\mathcal{B}}_{{\mathcal{L}}^2\left({\mathbb{R}}^2\right)}$, we derive

$$\begin{array}{c}\hfill {\Phi }_n{⊛}_{\mathbf{A}}{\mu }_m={\Phi }_m{⊛}_{\mathbf{A}}{\mu }_n\forall m,n\in \mathbb{N}.\end{array}$$

This implies that

$$\begin{array}{c}\hfill {\mathbf{O}}_{\psi }^{\mathbf{A}}\left({\mathrm{f}}_n{\ast }_{\mathbf{A}}{\mu }_m\right)={\mathbf{O}}_{\psi }^{\mathbf{A}}\left({\mathrm{f}}_m{\ast }_{\mathbf{A}}{\mu }_n\right)\forall m,n\in \mathbb{N}.\end{array}$$

Due to the fact that ${\mathbf{O}}_{\psi }^{\mathbf{A}}:L^2\left(\mathbb{R}\right)\to {\mathcal{L}}^2\left({\mathbb{R}}^2\right)$ is injective, we obtain

$${\mathrm{f}}_n{\ast }_{\mathbf{A}}{\mu }_m={\mathrm{f}}_m{\ast }_{\mathbf{A}}{\mu }_n\forall m,n\in \mathbb{N}.$$

Thus, $\left[\left({\mathrm{f}}_n\right)/\left({\mu }_n\right)\right]\in {\mathcal{B}}_{\mathbb{R}}^2$ and ${\mathfrak{O}}_{\psi }^{\mathbf{A}}\left[{\mathrm{f}}_n/{\mu }_n\right]=\left[\left({\mathbf{O}}_{\psi }^{\mathbf{A}}{\mathrm{f}}_n\right)/\left({\mu }_n\right)\right]=\left[\left({\Phi }_n\right)/\left({\mu }_n\right)\right]$. □

Theorem 6 

(Generalized convolution theorem). If  $X\in {\mathcal{B}}_{L^2\left(\mathbb{R}\right)}$ and $\mu \in \mathcal{D}\left(\mathbb{R}\right)$, then ${\mathfrak{O}}_{\psi }^{\mathbf{A}}(X{\ast }_{\mathbf{A}}\mu )={\mathfrak{O}}_{\psi }^{\mathbf{A}}X{⊛}_{\mathbf{A}}\mu$.

Proof. 

The demonstration that supports this theorem is clearly deduced from Theorem 1. □

Theorem 7. 

The extended OLCST ${\mathfrak{O}}_{\psi }^{\mathbf{A}}:{\mathcal{B}}_{L^2\left(\mathbb{R}\right)}\to {\mathcal{B}}_{{\mathcal{L}}^2\left({\mathbb{R}}^2\right)}$ maintains continuity associated with both δ-convergence and Δ-convergence.

Proof. 

Suppose $X_n\stackrel{\delta }{\to }X$ as $n\to \infty$ in ${\mathcal{B}}_{L^2\left(\mathbb{R}\right)}$. Then, $\exists \left({\mu }_n\right)\in {\Delta }_0$ so that $X_n{\ast }_{\mathbf{A}}{\mu }_k,X{\ast }_{\mathbf{A}}{\mu }_k\in L^2\left(\mathbb{R}\right)$ and $X_n{\ast }_{\mathbf{A}}{\mu }_k\to X{\ast }_{\mathbf{A}}{\mu }_k$ as $n\to \infty$ in $L^2\left(\mathbb{R}\right)$ for each fixed $k\in \mathbb{N}$. Given the continuity of the OLCST ${\mathbf{O}}_{\psi }^{\mathbf{A}}:L^2\left(\mathbb{R}\right)\to {\mathcal{L}}^2\left({\mathbb{R}}^2\right)$, we deduce that

$${\mathbf{O}}_{\psi }^{\mathbf{A}}\left(X_n{\ast }_{\mathbf{A}}{\mu }_k\right)⟶{\mathbf{O}}_{\psi }^{\mathbf{A}}\left(X{\ast }_{\mathbf{A}}{\mu }_k\right)\mathrm{as}n⟶\infty \mathrm{in}{\mathcal{L}}^2\left({\mathbb{R}}^2\right),$$

for every fixed $k\in \mathbb{N}$. By employing Theorems 2 and 6, we obtain

$${\mathfrak{O}}_{\psi }^{\mathbf{A}}{X}_n{\ast }_{\mathbf{A}}{\mu }_k={\mathfrak{O}}_{\psi }^{\mathbf{A}}\left(X_n{\ast }_{\mathbf{A}}{\mu }_k\right)={\mathbf{O}}_{\psi }^{\mathbf{A}}\left(X_n{\ast }_{\mathbf{A}}{\mu }_k\right),$$

which tends to

$${\mathbf{O}}_{\psi }^{\mathbf{A}}\left(X{\ast }_{\mathbf{A}}{\mu }_k\right)={\mathfrak{O}}_{\psi }^{\mathbf{A}}\left(X{\ast }_{\mathbf{A}}{\mu }_k\right)={\mathfrak{O}}_{\psi }^{\mathbf{A}}X{⊛}_{\mathbf{A}}{\mu }_k,$$

as $n\to \infty$ in ${\mathcal{L}}^2\left({\mathbb{R}}^2\right)$, for $k\in \mathbb{N}$. Therefore, ${\mathfrak{O}}_{\psi }^{\mathbf{A}}{X}_n\stackrel{\delta }{\to }{\mathfrak{O}}_{\psi }^{\mathbf{A}}X$ as $n\to \infty$.

Next, let us assume that $X_n\stackrel{\Delta }{\to }X$ as $n\to \infty$. Consequently, $\exists \left({\mu }_n\right)\in {\Delta }_0$ such that $\left(X_n-X\right){\ast }_{\mathbf{A}}$${\mu }_n\in L^2\left(\mathbb{R}\right)$ for all $n\in \mathbb{N}$, and $\left(X_n-X\right){\ast }_{\mathbf{A}}{\mu }_n$ converges to 0 as $n\to \infty$ in $L^2\left(\mathbb{R}\right)$. By leveraging Theorems 2, 3 and 6, we have

$$\begin{array}{c}\hfill \left({\mathfrak{O}}_{\psi }^{\mathbf{A}}{X}_n-{\mathfrak{O}}_{\psi }^{\mathbf{A}}X\right){⊛}_{\mathbf{A}}{\mu }_n={\mathfrak{O}}_{\psi }^{\mathbf{A}}\left[\left(X_n-X\right){\ast }_{\mathbf{A}}{\mu }_n\right]={\mathbf{O}}_{\psi }^{\mathbf{A}}\left[\left(X_n-X\right){\ast }_{\mathbf{A}}{\mu }_n\right]\forall n\in \mathbb{N}.\end{array}$$

Additionally, utilizing the continuity of ${\mathbf{O}}_{\psi }^{\mathbf{A}}$ on $L^2\left(\mathbb{R}\right)$, we obtain

$$\begin{array}{c}\hfill \left({\mathfrak{O}}_{\psi }^{\mathbf{A}}{X}_n-{\mathfrak{O}}_{\psi }^{\mathbf{A}}X\right){⊛}_{\mathbf{A}}{\mu }_n⟶0\mathrm{as}n⟶\infty \mathrm{in}{\mathcal{L}}^2\left({\mathbb{R}}^2\right).\end{array}$$

This shows that ${\mathfrak{O}}_{\psi }^{\mathbf{A}}{X}_n\stackrel{\Delta }{\to }{\mathfrak{O}}_{\psi }^{\mathbf{A}}X$ as $n\to \infty$. □

4. Discrete OLCST

The development of discrete OLCST relies on discretizing the parameters a and b. Mathematically, we achieve this by discretizing the continuous parameters a and b in Equation (3) as integral values. To effectively discretize the window functions, we opt for $a=a_0^m$ and $b=n{b}_0{a}_0^{-m}$, where $a_0$ and $b_0$ are constant positive values. This chosen approach results in a well-defined family of discretized window functions. Embracing this methodology not only simplifies the procedure but also enhances the accuracy of our analysis. Therefore, the set of discretized window functions is given as

$$\begin{array}{c}\hfill {\psi }_{m,n}^{\mathbf{A}}\left(t\right)=a_0^m{e}^{{\displaystyle \frac{j}{2B}}(A{n}^2{b}_0^2{a}_0^{-2m}+2\tau n{b}_0{a}_0^{-m})}\psi (a_0^{m}t-n{b}_0))e^{j{a}_0^{m}t-{\displaystyle \frac{j}{2B}}(A{t}^2+2\tau t)}.\end{array}$$

(7)

In this context, the parameters m and n play pivotal roles in dilation and translation, respectively. To simplify computations, we can set $a_0=2$ and $b_0=1$, resulting in a binary dilation of $2^m$ and a dyadic translation of $n{2}^{-m}$. With these choices, the simplified form of the discretized family (7) is

$$\begin{array}{c}\hfill {\psi }_{m,n}^{\mathbf{A}}\left(t\right)=2^m{e}^{{\displaystyle \frac{j}{2B}}(A{n}^2{2}^{-2m}+2\tau n{2}^{-m})}\psi (2^{m}t-n))e^{j{2}^{m}t-{\displaystyle \frac{j}{2B}}(A{t}^2+2\tau t)}.\end{array}$$

(8)

The precise definition of discrete OLCST is provided as follows.

Definition 7. 

For $\psi \in L^2\left(\mathbb{R}\right)$ and $\mathrm{f}\in L^2\left(\mathbb{R}\right)$, the discrete OLCST of f is defined by

$$\begin{array}{cc}\hfill {\mathbf{O}}_{\psi }^{\mathbf{A}}\left[\mathrm{f}\right](m,n)& =\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle \hfill \\ \hfill & ={\int }_{-\infty }^{\infty }\mathrm{f}\left(t\right)\overline{{\psi }_{m,n}^{\mathbf{A}}\left(t\right)}dt,m,n\in \mathbb{Z},\hfill \end{array}$$

(9)

where ${\psi }_{m,n}^{\mathbf{A}}\left(t\right)$ is given by (8).

Now, let us consider an example to illustrate discrete OLCST better.

Example 1. 

Assume the Gaussian function of unit amplitude as $\mathrm{f}\left(\mathrm{x}\right)=e^{-k{\mathrm{x}}^2}$, where $k>0$. The discrete OLCST of f concerning the rectangular function $\psi \left(\mathrm{x}\right)$ with a bounded width T is given by

$$\psi \left(\mathrm{x}\right)=\{\begin{array}{l}1,-{\displaystyle \frac{\mathrm{T}}{2}}<\mathrm{x}\le {\displaystyle \frac{\mathrm{T}}{2}},\\ 0,\mathit{elsewhere}.\end{array}$$

Using integer translations and dyadic dilations of ψ, we have

$$\psi (2^m\mathrm{x}-n)=\left\{\begin{array}{l}1,{\displaystyle \frac{1}{2^m}}\left(n-{\displaystyle \frac{\mathrm{T}}{2}}\right)<\mathrm{x}\le {\displaystyle \frac{1}{2^m}}\left(n+{\displaystyle \frac{\mathrm{T}}{2}}\right),\hfill \\ 0,\mathit{elsewhere}.\hfill \end{array}\right.$$

Therefore, the discrete OLCST is computed for f(x) as

$$\begin{array}{cc}\hfill {\mathbf{O}}_{\psi }^{\mathbf{A}}\left[\mathrm{f}\right](m,n)& =2^m{e}^{-{\displaystyle \frac{j}{2B}}(A{n}^2{2}^{-2m}+2\tau n{2}^{-m})}{\int }_{-\infty }^{\infty }\mathrm{f}\left(\mathrm{x}\right)\overline{\psi (2^m\mathrm{x}-n))}{e}^{-j{2}^m\mathrm{x}+{\displaystyle \frac{j}{2B}}(A{\mathrm{x}}^2+2\tau \mathrm{x})}d\mathrm{x}\hfill \\ \hfill & =2^m{e}^{-{\displaystyle \frac{j}{2B}}(A{n}^2{2}^{-2m}+2\tau n{2}^{-m})}{\int }_{2^{-m}(n-{\displaystyle \frac{\mathrm{T}}{2}})}^{2^{-m}(n+{\displaystyle \frac{T}{2}})}{e}^{({\displaystyle \frac{jA}{2B}}-k^2){\mathrm{x}}^2+({\displaystyle \frac{j\tau }{B}}-j{2}^m)\mathrm{x}}d\mathrm{x}\hfill \\ \hfill & =2^m{e}^{-{\displaystyle \frac{j}{2B}}(A{n}^2{2}^{-2m}+2\tau n{2}^{-m})}\left({\displaystyle \frac{i\sqrt{B}{e}^{-\frac{{\left(\tau -2^{m}B\right)}^2}{2B\left(2B{k}^2-iA\right)}}\sqrt{\pi }}{\sqrt{4B{k}^2-2iA}}}\right)\hfill \\ \hfill & \times (\mathit{erfi}\left({\displaystyle \frac{2^{-m-\frac{3}{2}}\left(2An+2^{m+1}\tau -2B\left(4^m-i{k}^2(2n-\mathrm{T})\right)-A\mathrm{T}\right)}{\sqrt{B}\sqrt{2B{k}^2-iA}}}\right)\hfill \\ \hfill & -\mathit{erfi}\left({\displaystyle \frac{2^{-m-\frac{3}{2}}\left(2^{m+1}\tau +A(2n+\mathrm{T})-2B\left(4^m-i{k}^2(2n+\mathrm{T})\right)\right)}{\sqrt{B}\sqrt{2B{k}^2-iA}}}\right)).\hfill \end{array}$$

Here, ‘erfi(·)’ represents the imaginary error function.

After formulating discrete OLCST, our primary objective is to reconstruct the original signal from the transformed one. The inversion process involves working with coefficients in the form of ${\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle }_{m,n\in \mathbb{Z}}$. This reconstruction follows two fundamental principles [47]: characterizing random functions from $L^2\left(\mathbb{R}\right)$ through the sequences from ${\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle }_{m,n\in \mathbb{Z}}$, and ensuring this characterization by numerical stability.

It is important to note that functions can be uniquely identified by their coefficients $\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle$. Specifically, if ${\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle }_{m,n\in \mathbb{Z}}={\langle \mathrm{g},{\psi }_{m,n}^{\mathbf{A}}\rangle }_{m,n\in \mathbb{Z}}$, then $\mathrm{f}=\mathrm{g}$. Likewise, if ${\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle }_{m,n\in \mathbb{Z}}=0$, then f = 0. Achieving this relies on ensuring that whenever f and g are close, the coefficient sequences ${\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle }_{m,n\in \mathbb{Z}}$ and ${\langle \mathrm{g},{\psi }_{m,n}^{\mathbf{A}}\rangle }_{m,n\in \mathbb{Z}}$ are close. In this context, closeness is defined using the standard l²-norm in the space $l^2\left({\mathbb{Z}}^2\right)$. The condition holds for a constant $\mathcal{Y}>0$, which is free from f, so that

$$\parallel {\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle }_{m,n\in \mathbb{Z}}{\parallel }^2=\sum _{m\in \mathbb{Z}}\sum _{n\in \mathbb{Z}}|\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle {|}^2\le \mathcal{Y}{\parallel \mathrm{f}\parallel }^2.$$

The practical interpretation of numerical stability in characterizing functions in $L^2\left(\mathbb{R}\right)$ through their sequences of coefficients is as follows: when the coefficient sequences of two functions are close in $l^2\left({\mathbb{Z}}^2\right)$, it indicates that they are close in $L^2\left(\mathbb{R}\right)$. Hence, if ${\sum }_{m\in \mathbb{Z}}{\sum }_{n\in \mathbb{Z}}{\left|\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle \right|}^2$ is lower, then ${\parallel \mathrm{f}\parallel }^2$ is also lower. More precisely, there should be a finite constant $\varpi <\infty$, such that if

$$\sum _{m\in \mathbb{Z}}\sum _{n\in \mathbb{Z}}{\left|\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle \right|}^2\le 1,$$

it implies that

$${\parallel \mathrm{f}\parallel }^2\le \varpi .$$

In simpler terms, when the sum of squared coefficients is bounded, it indicates that the norm of the original function is also bounded. This concept ensures stability in the numerical representation of functions in $L^2\left(\mathbb{R}\right)$. For $\mathrm{f}\in L^2\left(\mathbb{R}\right),\tilde{\mathrm{f}}=\mathrm{f}\left\{{\sum }_{m\in \mathbb{Z}}{\sum }_{n\in \mathbb{Z}}\right|\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle {|}^2{\}}^{-\frac{1}{2}}$. Subsequently, we can easily confirm that ${\sum }_{m\in \mathbb{Z}}{\sum }_{n\in \mathbb{Z}}|\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle {|}^2\le 1$, ensuring that ${\parallel \tilde{\mathrm{f}}\parallel }_2^2\le \varpi$. This indicates that the squared norm of the transformed signal, denoted as $\tilde{\mathrm{f}}$, is bounded by ϖ. Hence,

$${\left\{\sum _{m\in \mathbb{Z}}\sum _{n\in \mathbb{Z}}{\left|\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle \right|}^2\right\}}^{-1}{\parallel \mathrm{f}\parallel }^2\le \varpi \mathrm{or}\mathcal{X}{\parallel \mathrm{f}\parallel }^2\le \sum _{m\in \mathbb{Z}}\sum _{n\in \mathbb{Z}}{\left|\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle \right|}^2,$$

For a certain positive constant $\mathcal{X}={\varpi }^{-1}>0$, then the numerical stability of characterizing $\mathrm{f}\in L^2\left(\mathbb{R}\right)$ in terms of ${\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle }_{m,n\in \mathbb{Z}}$ is satisfied. This condition stipulates the existence of constants $0<\mathcal{X}\le \mathcal{Y}<\infty$, such that

$$\mathcal{X}{\parallel \mathrm{f}\parallel }^2\le \sum _{m\in \mathbb{Z}}\sum _{n\in \mathbb{Z}}|\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle {|}^2{\le \mathcal{Y}\parallel \mathrm{f}\parallel }^2.$$

(10)

Now, let us introduce the inversion formula for (9).

Theorem 8. 

Consider $\psi \in L^2\left(\mathbb{R}\right)$, the discrete OLCST of  $\mathrm{f}\left(t\right)$, as defined by (9). Assuming the stability condition (10), let us define linear operator T as

$$T(\mathrm{f})=\sum _{m\in \mathbb{Z}}\sum _{n\in \mathbb{Z}}\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle {\psi }_{m,n}^{\mathbf{A}},$$

(11)

and

$$\mathrm{f}\left(t\right)=\sum _{m\in \mathbb{Z}}\sum _{n\in \mathbb{Z}}\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle T^{-1}\left({\psi }_{m,n}^{\mathbf{A}}\right)\mathit{for}\mathrm{f}\in L^2\left(\mathbb{R}\right).$$

Proof. 

Condition (10) makes it easy to verify that T is a bounded linear operator with injective properties. This implies that for g = T(f), the following relationship holds:

$$\begin{array}{cc}\langle \mathrm{g},\mathrm{f}\rangle & =〈{\displaystyle \sum _{m\in \mathbb{Z}}}{\displaystyle \sum _{n\in \mathbb{Z}}}\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle {\psi }_{m,n}^{\mathbf{A}},\mathrm{f}〉\hfill \\ \hfill & \le {\mathcal{Y}\parallel f\parallel }^2.\hfill \end{array}$$

By employing the Cauchy–Schwarz property, we obtain

$$\begin{array}{cc}\hfill \mathcal{X}\parallel T^{-1}{\left(\mathrm{g}\right)\parallel }^2& =\mathcal{X}\parallel T^1{\left(T\left(\mathrm{f}\right)\right)\parallel }^2\hfill \\ \hfill & \le \sum _{m\in \mathbb{Z}}\sum _{n\in \mathbb{Z}}{\left|\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle \right|}^2\hfill \\ \hfill & =\parallel \mathrm{g}\parallel \parallel T^{-1}\left(\mathrm{g}\right)\parallel .\hfill \end{array}$$

Hence,

$$\parallel T^{-1}\left(\mathrm{g}\right)\parallel \le {\displaystyle \frac{1}{\mathcal{X}}}\parallel \mathrm{g}\parallel .$$

Therefore,

$$\mathrm{f}=T^{-1}T\left(\mathrm{f}\right)=\sum _{m\in \mathbb{Z}}\sum _{n\in \mathbb{Z}}\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle T^{-1}\left({\psi }_{m,n}^{\mathbf{A}}\right).$$

The proof of Theorem 8 is now complete. □

5. OLCST for Almost Periodic Functions (APFs)

In this section, we aim to broaden the application scope of OLCST to encompass persistent signals, with a particular emphasis on APFs. APFs, extending the concept of traditional periodic functions to the real line, have been extensively investigated across various mathematical, physical, and engineering disciplines [48,49,50]. Recent research efforts have focused on representing APFs using WFT [51], WT [52], FrFT [53], Gabor transform [54], etc. Now, we will try to extend the OLCST introduced in Equation (2) to the realm of APFs. This extension holds the potential to unveil new insights and enhance our understanding of these persistent signals.

Let the set of trigonometric polynomials (TPs) ${\mathfrak{T}}_P$ defined on $\mathbb{R}$ be given by

$$\begin{array}{c}\hfill {\mathfrak{T}}_P=\left[\mathrm{f}\left(t\right)=\sum _{k=1}^N{h}_k{e}^{i{\omega }_{k}t}:{\omega }_k\in \mathbb{R},h_k\in \mathbb{C},,N\in \mathbb{N}\right].\end{array}$$

where the norm is ${\parallel \mathrm{f}\parallel }_{\infty }={\sup }_{\mathrm{t}\in \mathbb{R}}\left|\mathrm{f}\right(\mathrm{t}\left)\right|$. By completing ${\mathfrak{T}}_P$ with this norm, we arrive at the space of APFs, denoted as ${\mathcal{A}}_P$. Alternatively, ${\mathcal{A}}_P$ can be regarded as the closed subspace of $L^{\infty }\left(\mathbb{R}\right)$, comprising functions of form $e^{i\omega t}$, where $\omega \in \mathbb{R}$. It is noteworthy that all APFs exhibit uniform continuity and boundedness.

An alternative norm on a set of TPs ${\mathfrak{T}}_P$ is established as

$$\begin{array}{c}\hfill {\parallel \mathrm{f}\parallel }_{{\mathcal{A}}_P^2}^2=\underset{T\to \infty }{\lim }{\displaystyle \frac{1}{2T}}{\int }_{-T}^T{\left|\mathrm{f}\right(t\left)\right|}^{2}dt,\end{array}$$

(12)

with inner product relation $\langle \mathrm{f},\mathrm{g}\rangle ={\lim }_{T\to \infty }{\displaystyle \frac{1}{2T}}{\int }_{-T}^T\mathrm{f}\left(\mathrm{t}\right)\overline{\mathrm{g}\left(\mathrm{t}\right)}d\mathrm{t}.$ The set of TPs ${\mathfrak{T}}_P$ is characterized by the norm (12), and it is important to note that this space is neither complete nor separable. The Hilbert space ${\mathcal{A}}_P^2$ (where ${\mathcal{A}}_P$ serves as a subset of ${\mathcal{A}}_P^2$) of Besicovitch APFs is obtained by completing ${\mathfrak{T}}_P$ with respect to this norm [48]. It is worth emphasizing that all APFs display uniform continuity and boundedness when considered with the aforementioned norm.

Consider the set $\mathcal{Q}\left(\mathbb{R}\right)$ of functions $\mathrm{q}$ given as

$$\mathrm{q}\left(\mathrm{x}\right)=\left\{\begin{array}{cc}\sum _{l=1}^L{\omega }_l{\mathrm{x}}^{{\alpha }_l},\hfill & \mathrm{x}\ge 0,\hfill \\ -\sum _{l=1}^L{\omega }_l{(-\mathrm{x})}^{{\alpha }_l},\hfill & \mathrm{x}<0.\hfill \end{array}\right.$$

In this context, L represents a natural number, and ${\omega }_l$ ranges over real values for $l=1,2,\cdots ,L$, where ${\alpha }_1>{\alpha }_2>\cdots >{\alpha }_L>0$. Consider a generalized trigonometric polynomial (GTP) on $\mathbb{R}$ as $P\left(\mathrm{x}\right)={\sum }_{k=1}^N{h}_k{e}^{i{\mathrm{q}}_k\left(\mathrm{x}\right)},$ where ${\mathrm{q}}_k\left(\mathrm{x}\right)\in Q\left(\mathbb{R}\right)$ and $h_k\in \mathbb{C}$.

Definition 8. 

For every $ϵ>0$, a function $f\in \mathbb{R}$ possess a robust limit power if there exists a GTP ${\mathcal{P}}_{ϵ}$, so that

$$\begin{array}{c}\hfill \parallel f-{\mathcal{P}}_{ϵ}\parallel =\sup \left\{|f\left(\mathrm{t}\right)-{\mathcal{P}}_{ϵ}|:\mathrm{t}\in \mathbb{R}\right\}<ϵ.\end{array}$$

The the inner product of collection $\mathbf{G}\left(\mathbb{R}\right),$ possessing a robust limit power, is defined as

$$\begin{array}{c}\hfill {\langle f,g\rangle }_{\mathbf{G}}=\underset{T\to \infty }{\lim }{\displaystyle \frac{1}{2T}}{\int }_{-T}^{T}f\left(\mathrm{t}\right)\overline{g\left(\mathrm{t}\right)}d\mathrm{t}.\end{array}$$

Hence, it is evident that ${\mathcal{A}}_P$ is a subset of $\mathbf{G}\left(\mathbb{R}\right)$, and the norms ${\parallel \mathrm{f}\parallel }_{{\mathcal{A}}_P^2}$ and ${\parallel \mathrm{f}\parallel }_{\mathbf{G}}$ are equivalent and both $\mathbf{G}\left(\mathbb{R}\right)$ and ${\mathcal{A}}_P$ are closed subspaces of $L^{\infty }\left(\mathbb{R}\right)$ [55].

Now, we will explore OLCST (2) in the space of APFs.

Theorem 9. 

If f is an APF, then the OLCST of f concerning   $\mathbf{A}=\left(\begin{array}{ccc}A& B;& \tau \\ C& D;& \eta \end{array}\right)$ with $A,B\ne 0$ emerges as a function with a robust limit power.

Proof. 

Let us initiate the proof by considering the case where f is a TP; that is,

$$\mathrm{f}\left(\mathrm{t}\right)=\sum _{k=1}^N{h}_k{e}^{j{\omega }_k\mathrm{t}}.$$

The OLCST of f is

$$\begin{array}{cc}\hfill {\mathbf{O}}_{\mathbf{A}}\left[\mathrm{f}\right(t)]\left(\xi \right)& ={\int }_{\mathbb{R}}\mathrm{f}\left(t\right){\mathcal{K}}_{\mathbf{A}}(t,\xi )dt\hfill \\ \hfill & =\sum _{k=1}^N{h}_k{\int }_{\mathbb{R}}{\displaystyle \frac{e^{j{\omega }_{k}t}}{\sqrt{j2\pi B}}}{e}^{{\displaystyle \frac{j}{2B}}(A{t}^2+2t(\tau -\xi )-2\xi (D\tau -B\eta )+D({\xi }^2+{\tau }^2))}dt\hfill \\ \hfill & =\sum _{k=1}^N{\displaystyle \frac{h_k}{\sqrt{j2\pi B}}}{e}^{{\displaystyle \frac{j}{2B}}\{D({\xi }^2+{\tau }^2)-2\xi (D\tau -B\eta )\}}{\int }_{\mathbb{R}}{e}^{{\displaystyle \frac{j}{2B}}(A{t}^2+2t(\tau -\xi )+2B{\omega }_{k}t)}dt\hfill \\ \hfill & =\sum _{k=1}^N{\displaystyle \frac{h_k}{\sqrt{j2\pi B}}}{e}^{{\displaystyle \frac{j}{2B}}\{D({\xi }^2+{\tau }^2)-2\xi (D\tau -B\eta )\}}{\int }_{\mathbb{R}}{e}^{\left\{-{\displaystyle \frac{A{t}^2}{2jB}}+(j{\omega }_k+{\displaystyle \frac{(\tau -\xi )j}{B}})t\right\}}dt.\hfill \end{array}$$

The integral part adopts the following expression

$${\int }_{\mathbb{R}}{e}^{\left\{-{\displaystyle \frac{A{t}^2}{2jB}}+(j{\omega }_k+{\displaystyle \frac{(\tau -\xi )j}{B}})t\right\}}dt=\sqrt{{\displaystyle \frac{j2\pi B}{A}}}{e}^{\left\{-j{\displaystyle \frac{{((\xi -\tau )-B{\omega }_k)}^2}{2AB}}\right\}}.$$

Therefore, we have,

$$\begin{array}{cc}\hfill {\mathbf{O}}_{\mathbf{A}}\left[\mathrm{f}\right(t)]\left(\xi \right)& =\sum _{k=1}^N{\displaystyle \frac{h_k}{\sqrt{A}}}{e}^{{\displaystyle \frac{j}{2B}}\{D({\xi }^2+{\tau }^2)-2\xi (D\tau -B\eta )\}}{e}^{\left\{-j{\displaystyle \frac{{((\xi -\tau )-B{\omega }_k)}^2}{2AB}}\right\}}\hfill \\ \hfill & ={\displaystyle \frac{1}{\sqrt{A}}}\sum _{k=1}^N{h}_k{e}^{j{\displaystyle \frac{((AD({\xi }^2+{\tau }^2)-2\xi (D\tau -B\eta )A)-{((\xi -\tau )-B{\omega }_k)}^2)}{2AB}}}.\hfill \end{array}$$

This expression represents a GTP in ξ. Moreover, for an APF f, there is a sequence of TPs $\left\{{\mathrm{f}}_n\right\}$ such that it uniformly converges to f. Hence, it is enough to show that the limit of $\parallel {\mathrm{f}}_n-\mathrm{f}\parallel$ tends to 0 as n approaches infinity, and then the limit of ${\parallel {\mathbf{O}}_{\mathbf{A}}{\mathrm{f}}_n-{\mathbf{O}}_{\mathbf{A}}\mathrm{f}\parallel }_{\infty }$ also tends to 0.

We obtain

$$\begin{array}{cc}\hfill |{\mathbf{O}}_{\mathbf{A}}\left[{\mathrm{f}}_n\right]\left(\xi \right)-{\mathbf{O}}_{\mathbf{A}}\left[\mathrm{f}\right]\left(\xi \right)|& \le {\displaystyle \frac{1}{\sqrt{j2\pi B}}}{\int }_{\mathbb{R}}\left|{\mathrm{f}}_n\left(t\right)-\mathrm{f}\left(t\right)\right|dt\to 0.\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill \parallel {\mathbf{O}}_{\mathbf{A}}\left[{\mathrm{f}}_n\right]\left(\xi \right)-{\mathbf{O}}_{\mathbf{A}}{\left[\mathrm{f}\right]\left(\xi \right)\parallel }_{\infty }\to 0.\end{array}$$

The proof of Theorem 9 is now complete. □

The following theorem shows that the presumption $\psi \in L^1\left(\mathbb{R}\right)$ is sufficient to ensure that OLCST represents a function with a robust limit power.

Theorem 10. 

Consider a window function $\psi \in L^1\cup L^2\left(\mathbb{R}\right)$. Then, for any APF $\mathrm{f}$, the OLCST ${\mathbf{O}}_{\psi }^{\mathbf{A}}\left[\mathrm{f}\right](a,b)$ is characterized as a function with a robust limit power in b.

Proof. 

Let f(t) be a TP defined as

$$\begin{array}{c}\hfill \mathrm{f}\left(\mathrm{t}\right)=\sum _{k=1}^N{h}_k{e}^{j{\omega }_k\mathrm{t}}.\end{array}$$

Therefore, the OLCST of f is

$${\mathbf{O}}_{\psi }^{\mathbf{A}}\left[\mathrm{f}\right](a,b)=a\sum _{k=1}^N{h}_k{e}^{-{\displaystyle \frac{j}{2B}}(A{b}^2+2\tau b)}{\int }_{\mathbb{R}}\overline{\psi \left(a\right(t-b\left)\right)}{e}^{j({\omega }_k-a)\mathrm{t}+{\displaystyle \frac{j}{2B}}(A{\mathrm{t}}^2+2\tau \mathrm{t})}d\mathrm{t}.$$

With $a(\mathrm{t}-b)=z$, we have

$$\begin{array}{cc}\hfill {\mathbf{O}}_{\psi }^{\mathbf{A}}\left[\mathrm{f}\right](a,b)& =e^{-{\displaystyle \frac{j}{2B}}(A{b}^2+2\tau b)}\sum _{k=1}^N{h}_k{\int }_{\mathbb{R}}\overline{\psi \left(z\right)}{e}^{j({\omega }_k-a)(b+{\displaystyle \frac{z}{a}})+{\displaystyle \frac{j}{2B}}(A{(b+{\displaystyle \frac{z}{a}})}^2+2\tau (b+{\displaystyle \frac{z}{a}}))}dz\hfill \\ \hfill & =e^{-{\displaystyle \frac{j}{2B}}(A{b}^2+2\tau b)}\sum _{k=1}^N{e}^{j({\omega }_k-a)b}{h}_k{\int }_{\mathbb{R}}\overline{\psi \left(z\right)}{e}^{j({\displaystyle \frac{z{\omega }_k}{a}}-z)+{\displaystyle \frac{j}{2B}}{\displaystyle \frac{A{z}^2}{a^2}}+{\displaystyle \frac{j}{B}}{\displaystyle \frac{Abz}{a}}+{\displaystyle \frac{j\tau z}{Ba}}}dz\hfill \\ \hfill & =e^{-{\displaystyle \frac{j}{2B}}(A{b}^2+2\tau b)}\sum _{k=1}^N{e}^{j({\omega }_k-a)b}{h}_k{\int }_{\mathbb{R}}\overline{\psi \left(z\right)}{e}^{\left\{-jz+j{\displaystyle \frac{A{z}^2}{2B{a}^2}}+j{\displaystyle \frac{A{z}^2}{2B}}+{\displaystyle \frac{j}{B}}\tau z\right\}}\hfill \\ \hfill & \times e^{-{\displaystyle \frac{j}{2B}}\{A{z}^2+2z(\tau -\left({\displaystyle \frac{B{\omega }_k+Ab+\tau }{a}}\right))-2\left({\displaystyle \frac{B{\omega }_k+Ab+\tau }{a}}\right)(D\tau -B\eta )+D{\left({\displaystyle \frac{B{\omega }_k+Ab+\tau }{a}}\right)}^2+D{\tau }^2\}}\hfill \\ \hfill & \times e^{{\displaystyle \frac{-j}{2B}}\left\{\left({\displaystyle \frac{B{\omega }_k+Ab+\tau }{a}}\right)(D\tau -B\eta )-D{\left({\displaystyle \frac{B{\omega }_k+Ab+\tau }{a}}\right)}^2-D{\tau }^2\right\}}dz\hfill \\ \hfill & =\sqrt{j2\pi B}\sum _{k=1}^N{h}_k{e}^{j({\omega }_k-a)b+{\displaystyle \frac{-j}{2B}}\left\{\left({\displaystyle \frac{B{\omega }_k+Ab+\tau }{a}}\right)(D\tau -B\eta )-D{\left({\displaystyle \frac{B{\omega }_k+Ab+\tau }{a}}\right)}^2-D{\tau }^2\right\}}\overline{{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B{\omega }_k+Ab+\tau }{a}}\right)},\hfill \end{array}$$

and here $\Psi \left(z\right)=\psi \left(z\right)e^{jz-j{\displaystyle \frac{A}{2B}}\left({\displaystyle \frac{1}{a^2}}+1\right)z^2-j{\displaystyle \frac{\tau z}{B}}}.$

Hence, for a fixed a, we infer that the OLCST ${\mathbf{O}}^{\mathbf{A}}\psi \left[\mathrm{f}\right](a,b)$ represents a GTP in b. Furthermore, if f is an APF, a sequence of GTPs f_n converges uniformly to f. Therefore, it is enough to demonstrate that whenever the limit ${\Vert {\mathrm{f}}_n-\mathrm{f}\Vert }_{\infty }$ approaches 0, the limit of ${\Vert {\mathbf{O}}^{\mathbf{A}}\psi \left[{\mathrm{f}}_n\right](a,b)-{\mathbf{O}}^{\mathbf{A}}\psi \left[\mathrm{f}\right](a,b)\Vert }_{\infty }$ also tends to 0.

Also,

$$\begin{array}{cc}\hfill \parallel {\mathbf{O}}_{\psi }^{\mathbf{A}}\left[{\mathrm{f}}_n\right](a,b)-{\mathbf{O}}_{\psi }^{\mathbf{A}}{\left[\mathrm{f}\right](a,b)\parallel }_{\infty }& \le \parallel {\mathrm{f}}_n{-\mathrm{f}\parallel }_{\infty }{\parallel \psi \parallel }_1\to 0.\hfill \end{array}$$

Hence, we observed that the OLCST ${\mathbf{O}}_{\psi }^{\mathbf{A}}\left[\mathrm{f}\right](a,b)$ of an APF f represents a function with a robust limit power in b. □

Now, we will develop a frame decomposition for APFs using OLCST.

Theorem 11. 

For any APF $\mathrm{f},\exists$ invariables ${\mathfrak{C}}_1,{\mathfrak{C}}_2>0$ so that

$${\mathfrak{C}}_1{\parallel \mathrm{f}\parallel }_{{\mathcal{A}}_P^2}^2\le {\displaystyle \frac{1}{2N+1}}\sum _{n=-N}^N\sum _{m=-\infty }^{\infty }|\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle {|}^2\le {\mathfrak{C}}_2{\parallel \mathrm{f}\parallel }_{{\mathcal{A}}_P^2}^2.$$

(13)

Proof. 

For TP $\mathrm{f}\left(t\right)={\sum }_{k=1}^K{h}_k{e}^{j{\omega }_{k}t},$ we obtain

$$\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle =\sum _{k=1}^K{2}^m{h}_k{\int }_{\mathbb{R}}{e}^{{\displaystyle \frac{-j}{2B}}(A{n}^2{2}^{-2m}+2\tau n{2}^{-m})}\overline{\psi (2^{m}t-n))}{e}^{j({\omega }_k-2^m)t+{\displaystyle \frac{j}{2B}}(A{t}^2+2\tau t)}dt.$$

With $2^{m}t-n=z,$ we have

$$\begin{array}{cc}\hfill \langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle & =\sum _{k=1}^K{h}_k{\int }_{\mathbb{R}}\overline{\psi \left(z\right)}{e}^{\left\{j{2}^{-m}({\omega }_k-2^m)(n+z)+{\displaystyle \frac{j}{2B}}(A{(n+z)}^2{2}^{-2m}+2\tau (n+z)2^{-m})-{\displaystyle \frac{j}{2B}}(A{n}^2{2}^{-2m}+2\tau n{2}^{-m})\right\}}dz\hfill \\ \hfill & =\sum _{k=1}^K{h}_k{e}^{j\left({\displaystyle \frac{{\omega }_k}{2^m}}-1\right)n}{\int }_{\mathbb{R}}\overline{\psi \left(z\right)}{e}^{\left\{j{\displaystyle \frac{{\omega }_{k}z}{2^m}}-jz+{\displaystyle \frac{j}{2B}}(A{z}^2{2}^{-2m}+2Anz{2}^{-2m})+{\displaystyle \frac{j}{B}}\tau z{2}^{-m}\right\}}dz\hfill \\ \hfill & =\sum _{k=1}^K{h}_k{e}^{j\left({\displaystyle \frac{{\omega }_k}{2^m}}-1\right)n}{\int }_{\mathbb{R}}\overline{\psi \left(z\right)}{e}^{\left\{-jz+{\displaystyle \frac{j}{2B}}\left({\displaystyle \frac{A{z}^2}{2^{2m}}}+{\displaystyle \frac{2Anz}{2^{2m}}}+{\displaystyle \frac{2\tau z}{2^m}}+A{z}^2+2\tau z\right)\right\}}\hfill \\ \hfill & \times e^{-{\displaystyle \frac{j}{2B}}\{A{z}^2+2z(\tau -{\displaystyle \frac{B{\omega }_k}{2^m}})-2\left({\displaystyle \frac{B{\omega }_k}{2^m}}\right)(D\tau -B\eta )+D{\left({\displaystyle \frac{B{\omega }_k}{2^m}}\right)}^2+D{\tau }^2\}}{e}^{{\displaystyle \frac{-j}{2B}}\left\{\left({\displaystyle \frac{B{\omega }_k}{2^m}}\right)(D\tau -B\eta )-D{\left({\displaystyle \frac{B{\omega }_k}{2^m}}\right)}^2-D{\tau }^2\right\}}dz\hfill \\ \hfill & =\sum _{k=1}^K{h}_k{e}^{j\left({\displaystyle \frac{{\omega }_k}{2^m}}-1\right)n+{\displaystyle \frac{jD}{2B}}\left({\left({\displaystyle \frac{B{\omega }_k}{2^m}}\right)}^2+{\tau }^2\right)-j{\displaystyle \frac{{\omega }_k}{2^m}}(D\tau -B\eta )}{\int }_{\mathbb{R}}\overline{\psi \left(z\right)}{e}^{\left\{-jz+{\displaystyle \frac{j}{2B}}\left({\displaystyle \frac{A{z}^2}{2^{2m}}}+{\displaystyle \frac{2Anz}{2^{2m}}}+{\displaystyle \frac{2\tau z}{2^m}}+A{z}^2+2\tau z\right)\right\}}\overline{{\mathcal{K}}_{\mathbf{A}}(z,{\displaystyle \frac{B{\omega }_k}{2^m}})}dz\hfill \\ \hfill & =\sum _{k=1}^K{h}_k{e}^{j\left({\displaystyle \frac{{\omega }_k}{2^m}}-1\right)n+{\displaystyle \frac{jD}{2B}}\left({\left({\displaystyle \frac{B{\omega }_k}{2^m}}\right)}^2+{\tau }^2\right)-j{\displaystyle \frac{{\omega }_k}{2^m}}(D\tau -B\eta )}\overline{{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B{\omega }_k}{2^m}}\right)},\hfill \end{array}$$

where $\Psi \left(z\right)=\psi \left(z\right)e^{jz-{\displaystyle \frac{j}{2B}}\left({\displaystyle \frac{A{z}^2}{2^{2m}}}+{\displaystyle \frac{2Anz}{2^{2m}}}+{\displaystyle \frac{2\tau z}{2^m}}+A{z}^2+2\tau z\right)}.$

Moreover, we observe that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2N+1}}\sum _{n=-N}^N\sum _{m=-\infty }^{\infty }{\left|\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle \right|}^2\hfill \\ \hfill & ={\displaystyle \frac{\left|2\pi B\right|}{2N+1}}\sum _{n=-N}^N\sum _{m=-\infty }^{\infty }\sum _{k=1}^K\sum _{l=1}^K{h}_k\overline{h_l}{e}^{{\displaystyle \frac{j}{2^m}}({\omega }_k-{\omega }_l)n+{\displaystyle \frac{jD}{2B}}{\left({\displaystyle \frac{B}{2^m}}\right)}^2({\omega }_k^2-{\omega }_l^2)-j{\displaystyle \frac{({\omega }_k-{\omega }_l)}{2^m}}(D\tau -B\eta )}\overline{{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B{\omega }_k}{2^m}}\right)}{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B{\omega }_l}{2^m}}\right).\hfill \end{array}$$

(14)

As $N\to \infty ,$ Equation (14) simplifies to

$${\displaystyle \frac{1}{2N+1}}\sum _{n=-N}^N\sum _{m=-\infty }^{\infty }|\langle \mathrm{f},{\psi }_{m,n}^{\mathbf{A}}\rangle {|}^2=\left|2\pi B\right|\left\{\sum _{k=1}^K|h_k{|}^2\left\{\sum _{m=-\infty }^{\infty }{\left|{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B{\omega }_k}{2^m}}\right)\right|}^2\right\}+\mathrm{Rest}\left(\mathrm{f}\right)\right\},$$

Rest $\left(\mathrm{f}\right)=\sum h_k\overline{h_l}\Delta ({\omega }_k,{\omega }_l),$ and the summation includes all pairs $k,l$, where ${\omega }_k-{\omega }_l$ is a non-zero multiplication of $2^m$ and

$$\begin{array}{c}\hfill \Delta ({\omega }_k,{\omega }_l)=\sum _{m=-\infty }^{\infty }\overline{{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B{\omega }_k}{2^m}}\right)}{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B{\omega }_l}{2^m}}\right)e^{{\displaystyle \frac{jD}{2B}}{\left({\displaystyle \frac{B}{2^m}}\right)}^2({\omega }_k^2-{\omega }_l^2)-j{\displaystyle \frac{({\omega }_k-{\omega }_l)}{2^m}}(D\tau -B\eta )}.\end{array}$$

Now, for Rest (f), we apply the Cauchy–Schwarz property:

$$\begin{array}{cc}\hfill \left|\mathrm{Rest}\right(\mathrm{f}\left)\right|& =\left|\sum _{\omega \in \mathbb{R}}\sum _{s\in \mathbb{Z}-\left\{0\right\}}{h}_{\omega }\overline{h_{\omega +s{2}^m}}\sum _{m=-\infty }^{\infty }\overline{{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B\omega }{2^m}}\right)}{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B\omega }{2^m}}+Bs\right)e^{-{\displaystyle \frac{jD}{Bs}}2\left({\displaystyle \frac{2\omega }{2^m}}+s\right)}{e}^{js(D\tau -B\eta )}\right|\hfill \\ \hfill & \le \sum _{s\in \mathbb{Z}-\left\{0\right\}}{\left\{\sum _{m=-\infty }^{\infty }\sum _{\omega \in \mathbb{R}}{\left|h_{\omega }\right|}^2\left|{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B\omega }{2^m}}\right)\right|\left|{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B\omega }{2^m}}+Bs\right)\right|\right\}}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \times {\left\{\sum _{m=-\infty }^{\infty }\sum _{\omega \in \mathbb{R}}{\left|h_{\omega +s{2}^m}\right|}^2\left|{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B\omega }{2^m}}\right)\right|\left|{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B\omega }{2^m}}+Bs\right)\right|\right\}}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le \sum _{\omega \in \mathbb{R}}{\left|h_{\omega }\right|}^2\sum _{s\in \mathbb{Z}-\left\{0\right\}}{\left(\Gamma \left(s\right)\Gamma (-s)\right)}^{{\displaystyle \frac{1}{2}}},\hfill \end{array}$$

where $\Gamma \left(s\right)={\sum }_{\omega \in \mathbb{R}}\left\{{\sum }_{m=-\infty }^{\infty }\left|{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B\omega }{2^m}}\right)\right|\left|{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B\omega }{2^m}}+Bs\right)\right|\right\}.$

Similar to the assumptions formulated by Daubechies [47] when establishing wavelet frames within $L^2\left(\mathbb{R}\right)$, we will embrace the same hypothesis, that

$$\begin{array}{cc}\hfill {\mathfrak{C}}_1& =\underset{\omega \in \mathbb{R}}{\inf }\sum _{m=-\infty }^{\infty }{\left|{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B\omega }{2^m}}\right)\right|}^2-\sum _{s\in \mathbb{Z}-\left\{0\right\}}{\left(\Gamma \left(s\right)\Gamma (-s)\right)}^{{\displaystyle \frac{1}{2}}}>0,\hfill \\ \hfill {\mathfrak{C}}_2& =\underset{\omega \in \mathbb{R}}{\sup }\sum _{m=-\infty }^{\infty }{\left|{\mathbf{O}}_{\mathbf{A}}\left[\Psi \right]\left({\displaystyle \frac{B\omega }{2^m}}\right)\right|}^2+\sum _{s\in \mathbb{Z}-\left\{0\right\}}{\left(\Gamma \left(s\right)\Gamma (-s)\right)}^{{\displaystyle \frac{1}{2}}}<\infty ,\hfill \end{array}$$

This gives rise to the inequality (13), which is applicable to TPs. Through the use of a standard approximation argument, we can infer that the same conclusion holds for APFs. □

6. Potential Application

Let $L_p\left(\mathbb{R}\right)$ be the space of complex-valued Lebesgue integrable functions on real-line R and δ be the delta sequence. Then, the equivalence class of the quotients is called the integrable Boehmian, the space of which is denoted by ${\mathcal{B}}_{L^2\left(\mathbb{R}\right)}.$ One may refer to Mikusinski [56], where Fourier transform for integrable Boehmians is investigated. By using the relation between OLCT and OLCST, one can define and find OLCST for integrable Boehmians.

This OLCST model features a greater number of free parameters than the existing FrST [27,57] and LCST [33,34] models. This increased flexibility enables the OLCST to outperform in the time–frequency analysis of chirp signals. The enhanced performance is demonstrated through the high-resolution spectrum achieved by the transform, the effective detection of disturbed chirp signals, and the representation of the linear frequency modulation (LFM) signal. The LFM signal is a critical non-stationary signal widely used in information systems such as sonar, radar, and communications. Moreover, OLCST can provide an optimal time–frequency distribution with a high time–frequency concentration.

7. Conclusions

In conclusion, this article has employed the convolution theorem to establish a B-S encompassing the OLCST of all square-integrable Boehmians. Through rigorous proof, we have demonstrated that the extended OLCST on square-integrable Boehmians is consistent with the traditional OLCST. Moreover, we have established key properties, affirming linearity, injectiveness, and continuity concerning both δ-convergence and Δ-convergence. This study contributes to the understanding of extended OLCST, providing a foundation for further exploration and applications in the realm of B-Ss. Also, we have introduced a discrete iteration of the OLCST. Furthermore, the breadth of our proposed work is expanded as we delve into the study of the OLCST within the domain of APFs. This extension underscores the versatility and applicability of the proposed methodology, offering valuable insights into its potential applications across various domains. The effectiveness and performance of the proposed transform can be further evaluated through simulation analyses of OLCST in future research.