Convolution and Sampling Theorems for Offset Fractional Fourier Transform

Abstract

The offset fractional Fourier transform (OFrFT) has emerged as a mathematical tool for time-frequency analysis of non-stationary signals. However, there has been relatively little research on investigating properties including the convolution theorem and the sampling formulas. This paper investigates the new form of convolution theorem and its product theorem associated with the OFrFT. We also establish a sampling formula related to the transformation. Finally, a simple example is displayed to verify the results related to the sampling formula for the OFrFT.

1. Introduction

It is well known that convolution is a fundamental concept in the theory of linear time-invariant (LTI) systems [1]. The convolution of an input signal with the impulse response of an LTI system produces the system output. The impulse response characterizes the system’s behavior, and convolution incorporates this behavior into the input signal to generate the output. In addition to this, convolution is closely related to the classical Fourier transform (FT) through the convolution theorem, which states that the FT of a convolution of two signals equals the product of their FTs. Building on this relationship, many authors have developed the convolution theorem in different types of transformations. For instance, in [2,3], the authors explored the convolution theorem for the fractional Fourier transform (FrFT). As is known, the offset fractional Fourier transform (OFrFT) is a general version of the FrFT, so the properties of the OFrFT are an extension of the corresponding properties of the FrFT [4,5,6,7,8,9,10,11,12,13]. Especially, the authors of [14] presented a convolution theorem for the OFrFT, which involves the multiplication of their OFrFT and a phase factor. The form of this convolution theorem contains an additional extra chirp factor and is not exactly similar to the convolution theorem for the FT. Compared to the convolution theorem proposed in [14], our result discussed in this work is simpler and closer to the convolution theorem related to the FT in the existing literature [15]. Additionally, the product theorem related to the proposed convolution definition is also established in detail.

On the other hand, the theory of the sampling formula associated with the FT was studied by the author in [16] and its application to time-varying systems was also demonstrated in [17]. A variant of the sampling formula is a generalized exponential sampling operator, which was recently studied by the authors of [18,19,20]. Further, the sampling formula for the FT was expanded in the framework of the FrFT [21,22,23]. As a non-trivial generalization of the FrFT, there is an inevitable desire to expand the sampling formula to the OFrFT domains. Therefore, the present research also aims to investigate the sampling formula related to this transformation, which has not appeared in the literature so far.

The organization of the work is as follows. In Section 2, we recall some essential facts on the FrFT, the OFrFT, and their basic properties. Section 3 focuses on discussing the fundamental properties of OFrFT that have not been studied in the existing literature. These properties will be very useful in solving partial differential equations in the OFrFT setting. Section 4 is devoted to the derivation of convolution and its product theorems concerning the OFrFT. The derivation of the sampling formula associated with the OFrFT is discussed in Section 5. Additionally, we include a simple example illustration related to the sampling formula. The paper ends with the conclusion in Section 6.

2. Preliminaries

In this part, we provide a brief review on some results that will be useful in the next section. We begin by introducing the following notations:

Definition 1.

For $1\le s<\infty$, the space of measurable functions f defined on $\mathbb{R}$ denoted by $L^s\left(\mathbb{R}\right)$, such that

$${\parallel f\parallel }_{L^s\left(\mathbb{R}\right)}={\left({\int }_{\mathbb{R}}{\left|f\left(t\right)\right|}^{s}dt\right)}^{{\displaystyle \frac{1}{s}}}<\infty .$$

(1)

Especially, for $s\to \infty$, we get

$$\begin{array}{c}\hfill {\parallel f\parallel }_{L^{\infty }\left(\mathbb{R}\right)}={\mathrm{ess}sup}_{t\in \mathbb{R}}\left|f\left(t\right)\right|.\end{array}$$

(2)

The usual inner product of $L^2\left(\mathbb{R}\right)$ is then defined as

$$\langle f,g\rangle ={\int }_{\mathbb{R}}f\left(t\right)\overline{g\left(t\right)}dt,$$

(3)

where overline means the conjugation.

In the following, we recall the definition of the fractional Fourier transform (FrFT) and its inverse, see, e.g., [2,3,24].

Definition 2.

The FrFT of any function f belongs to $L^2\left(\mathbb{R}\right)$ with the real parameter β defined by

$${\mathcal{F}}_{\beta }\left\{f\right\}\left(\eta \right)={\int }_{\mathbb{R}}f\left(t\right){\mathcal{K}}_{\beta }(t,\eta )dt,$$

(4)

where the kernel ${\mathcal{K}}_{\beta }(t,\eta )$ is

$$\begin{array}{c}\hfill {\mathcal{K}}_{\beta }(t,\eta )=A_{\beta }{e}^{i(t^2+{\eta }^2){\displaystyle \frac{cot\beta }{2}}-it\eta csc\beta },\beta \ne n\pi ,n\in \mathbb{Z},\end{array}$$

(5)

and

$$\begin{array}{c}\hfill A_{\beta }=\sqrt{{\displaystyle \frac{1-icot\beta }{2\pi }}}={\displaystyle \frac{e^{i\beta /2}}{\sqrt{2i\pi sin\beta }}}.\end{array}$$

(6)

We describe the basic properties of the kernel ${\mathcal{K}}_{\beta }(t,\eta )$:

$$\begin{array}{c}\hfill {\mathcal{K}}_{\beta }(t,\eta )=\delta (t-\eta ),\mathrm{for}\beta =2n\pi ,\end{array}$$

(7)

and

$$\begin{array}{c}\hfill {\mathcal{K}}_{\beta }(t,\eta )=\delta (t+\eta ),\mathrm{for}\beta =(2n+1)\pi ,n\in \mathbb{Z},\end{array}$$

(8)

where $\delta (t-\eta )$ is Dirac’s delta.

Definition 3.

Assume that any function f and its FrFT ${\mathcal{F}}_{\beta }\left\{f\right\}$ are in $L^1\left(\mathbb{R}\right)$. The inversion formula for the FrFT is calculated by

$$\begin{array}{c}\hfill f\left(t\right)={\mathcal{F}}_{\beta }^{-1}\left[{\mathcal{F}}_{\beta }\left\{f\right\}\right]\left(t\right)={\int }_{\mathbb{R}}{\mathcal{F}}_{\beta }\left\{f\right\}\left(\eta \right)\overline{A_{\beta }}{e}^{-i(t^2+{\eta }^2){\displaystyle \frac{cot\beta }{2}}+it\eta csc \beta }d\eta .\end{array}$$

(9)

The direct interaction between the FrFT and the FT is

$$\begin{array}{c}\hfill {\mathcal{F}}_{\beta }\left\{f\right\}\left(\eta \right)=A_{\beta }{e}^{i{\eta }^2{\displaystyle \frac{cot\beta }{2}}}\mathcal{F}\left\{\check{f}\right\}(\eta csc\beta ),\end{array}$$

(10)

for which

$$\begin{array}{c}\hfill \check{f}\left(t\right)=f\left(t\right)e^{i{t}^2{\displaystyle \frac{cot\beta }{2}}}.\end{array}$$

(11)

Here, the definition of the Fourier transform (FT) for signal f in $L^2\left(\mathbb{R}\right)$ is described by (see [15,25,26,27])

$$\mathcal{F}\left\{f\right\}\left(\eta \right)=\widehat{f}\left(\eta \right)={\int }_{\mathbb{R}}f\left(t\right)e^{-it\eta }dt,$$

(12)

and its inverse is

$$f\left(t\right)={\displaystyle \frac{1}{2\pi }}{\int }_{\mathbb{R}}\mathcal{F}\left\{f\right\}\left(\eta \right)e^{it\eta }d\eta .$$

(13)

The subsequent definition refers to the OFrFT [14] mentioned earlier, which may be interpreted as a natural expansion of the FrFT.

Definition 4.

The definition of the OFrFT for function f belonging to $L^2\left(\mathbb{R}\right)$ is expressed as

$$\begin{array}{c}\hfill {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)={\widehat{f}}_{\mathbb{O}}\left(\eta \right)={\int }_{\mathbb{R}}f\left(t\right){\mathcal{K}}_{(\beta ,m,n)}(t,\eta )dt,\end{array}$$

(14)

where

$${\mathcal{K}}_{(\beta ,m,n)}(t,\eta )=A_{\beta }{e}^{{\displaystyle \frac{i}{2}}\left(({\eta }^2+t^2+m^2)cot\beta +2t(m-\eta )csc\beta +2\eta (n-m cot \beta )\right)}.$$

(15)

Here, the parameters $(\beta ,m,n)$ are real numbers.

It is straightforward to verify that when $m=n=0$ in Equation (14), we obtain the definition of the FrFT (4).

The next lemma describes the natural relation between the OFrFT and the FT, which is central in deriving the upcoming results related to the proposed OFrFT.

Lemma 1.

For every function f belonging to $L^2\left(\mathbb{R}\right)$, the following relation is satisfied:

$$\begin{array}{c}\hfill \mathcal{F}\left\{f_{\beta }\right\}(\eta csc\beta )={\displaystyle \frac{1}{A_{\beta }}}{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)e^{-{\displaystyle \frac{i}{2}}(({\eta }^2+m^2)cot\beta +2\eta (n-m cot \beta ))},\end{array}$$

(16)

where

$$\begin{array}{c}\hfill f_{\beta }\left(t\right)=f\left(t\right)e^{{\displaystyle \frac{i}{2}}(t^{2}cot\beta +2tm csc \beta )}.\end{array}$$

(17)

Next, one can easily generate the original function f in terms of its ${\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}$ by utilizing the following definition.

Definition 5.

For any function f and its OFrFT ${\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}$ belonging to $L^1\left(\mathbb{R}\right)$, the inverse of the OFrFT is expressed as

$$f\left(t\right)={\int }_{\mathbb{R}}{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)\overline{{\mathcal{K}}_{(\beta ,m,n)}(t,\eta )}d\eta .$$

(18)

The Parseval’s identity related to the OFrFT defined by (14) is expressed in the following form:

Theorem 1.

Any two functions $f,g\in L^1\left(\mathbb{R}\right)\cap L^2\left(\mathbb{R}\right)$ are related to their OFrFT via Parseval’s identity as

$$\begin{array}{c}\hfill {\langle f,g\rangle }_{L^2\left(\mathbb{R}\right)}={〈{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\},{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{g\right\}〉}_{L^2\left(\mathbb{R}\right)},\end{array}$$

(19)

In particular,

$${\parallel f\parallel }_{L^2\left(\mathbb{R}\right)}^2={\parallel {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\parallel }_{L^2\left(\mathbb{R}\right)}^2.$$

(20)

Some key properties of the OFrFT compared to the FrFT were studied by the authors in [28].

3. Fundamental Properties for the OFrFT

In [28], the authors derived some properties of the OFrFT, such as shifting, modulation, and uncertainty principles. Now, let us establish the OFrFT of the nth derivative, for which the result was not published in [28].

Theorem 2.

Suppose that $f\left(t\right)$ is continuous n-times differentiable, $f,f^{\prime },\cdots ,f^{\left(r\right)}\in L^1\left(\mathbb{R}\right)$ and if we assume that

$$\begin{array}{c}\hfill \underset{\left|t\right|\to \infty }{lim}{f}^{\left(r\right)}\left(t\right)=0,r=0,1,2,\cdots ,n-1,\end{array}$$

then

$$\begin{array}{c}\hfill {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{{\displaystyle \frac{d^{r}f}{d{t}^r}}\right\}\left(\eta \right)={\left(i(\eta -m-n cot \beta )sin\beta +cos\beta {\displaystyle \frac{d}{d\eta }}\right)}^r{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right).\end{array}$$

(21)

Proof. 

For $r=1$, we have

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{{\displaystyle \frac{df}{dt}}\right\}\left(\eta \right)\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}{\displaystyle \frac{df}{dt}}{e}^{{\displaystyle \frac{i}{2}}\left((t^2+{\eta }^2+m^2)cot\beta +2t(m-\eta )csc\beta +2\eta (n-m cot \beta )\right)}dt\hfill \\ \hfill & =A_{\beta }(e^{{\displaystyle \frac{i}{2}}\left((t^2+{\eta }^2+m^2)cot\beta +2t(m-\eta )csc\beta +2\eta (n-m cot \beta )\right)}f\left(t\right){|}_{-\infty }^{\infty }\hfill \\ \hfill & -{\int }_{\mathbb{R}}f\left(t\right)\left(itcot\beta +i(m-\eta )csc\beta \right)e^{{\displaystyle \frac{i}{2}}\left((t^2+{\eta }^2+m^2)cot\beta +2t(m-\eta )csc\beta +2\eta (n-m cot \beta )\right)}dt)\hfill \\ \hfill & =-A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)\left(itcot\beta +i(m-\eta )csc\beta \right)e^{{\displaystyle \frac{i}{2}}\left((t^2+{\eta }^2+m^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta )\right)}dt.\hfill \end{array}$$

This equation is equal to

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{{\displaystyle \frac{df}{dt}}\right\}\left(\eta \right)\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)\left(i(\eta -m)csc\beta -itcot\beta \right)\hfill \\ \hfill & \times e^{{\displaystyle \frac{i}{2}}\left((t^2+{\eta }^2+m^2)cot\beta +2t(m-\eta )csc\beta +2\eta (n-m cot \beta )\right)}dt\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)\left(i(\eta -m)(sin\beta +cos\beta cot\beta )-itcot\beta \right)\hfill \\ \hfill & \times e^{{\displaystyle \frac{i}{2}}\left((t^2+{\eta }^2+m^2)cot\beta +2t(m-\eta )csc\beta +2\eta (n-m cot \beta )\right)}dt\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)\left(i(\eta -m)sin\beta +icos\beta \left(\right(\eta -m)cot\beta -tcsc\beta )\right)\hfill \\ \hfill & \times e^{{\displaystyle \frac{i}{2}}\left((t^2+{\eta }^2+m^2)cot\beta +2t(m-\eta )csc\beta +2\eta (n-m cot \beta )\right)}dt\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)\left(i(\eta -m)sin\beta +icos\beta \left(\right(\eta -m)cot\beta -tcsc\beta +in-in)\right)\hfill \\ \hfill & \times e^{{\displaystyle \frac{i}{2}}\left((t^2+{\eta }^2+m^2)cot\beta +2t(m-\eta )csc\beta +2\eta (n-m cot \beta )\right)}dt.\hfill \end{array}$$

After simplification, we get

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{{\displaystyle \frac{df}{dt}}\right\}\left(\eta \right)\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)\left(i(\eta -m)sin\beta \right)e^{{\displaystyle \frac{i}{2}}\left((t^2+{\eta }^2+m^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta )\right)}dt\hfill \\ \hfill & +A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)cos\beta (i\eta cot\beta -itcsc\beta -imcot\beta +in-in)\hfill \\ \hfill & \times e^{{\displaystyle \frac{i}{2}}\left((t^2+m^2+{\eta }^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta )\right)}dt\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)\left((i\eta -im)sin\beta -incos\beta \right)e^{{\displaystyle \frac{i}{2}}\left((t^2+m^2+{\eta }^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta )\right)}dt\hfill \\ \hfill & +A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)cos\beta (i\eta cot\beta -itcsc\beta -imcot\beta +in)\hfill \\ \hfill & \times e^{{\displaystyle \frac{i}{2}}\left((t^2+m^2+{\eta }^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta )\right)}dt.\hfill \end{array}$$

(22)

Therefore,

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{{\displaystyle \frac{df}{dt}}\right\}\left(\eta \right)\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)\left(i(\eta -m-ncot\beta )sin\beta \right)e^{{\displaystyle \frac{i}{2}}\left((t^2+m^2+{\eta }^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta )\right)}dt\hfill \\ \hfill & +A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)cos\beta (i\eta cot\beta -itcsc\beta -imcot\beta +in)\hfill \\ \hfill & \times e^{{\displaystyle \frac{i}{2}}\left((t^2+m^2+{\eta }^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta )\right)}dt\hfill \\ \hfill & =i(\eta -m-ncot\beta )sin\beta {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)+cos\beta {\displaystyle \frac{d{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}}{d\eta }}\left\{f\right\}\left(\eta \right)\hfill \\ \hfill & =\left(i(\eta -m-ncot\beta )sin\beta +cos\beta {\displaystyle \frac{d}{d\eta }}\right){\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right).\hfill \end{array}$$

(23)

Assume that Equation (23) is true for $r=k-1$, namely,

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{{\displaystyle \frac{d^{k-1}}{d{t}^{k-1}}}f\right\}\left(\eta \right)& ={\left(i(\eta -m-ncot\beta )sin\beta +cos\beta {\displaystyle \frac{d}{d\eta }}\right)}^{k-1}{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right).\hfill \end{array}$$

(24)

It follows that

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{{\displaystyle \frac{d^k}{d{t}^k}}f\right\}\left(\eta \right)\hfill \\ \hfill & ={\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{{\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{d^{k-1}}{d{t}^{k-1}}}\right)f\right\}\left(\eta \right)\hfill \\ \hfill & =\left(i(\eta -m-ncot\beta )sin\beta +cos\beta {\displaystyle \frac{d}{d\eta }}\right){\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{{\displaystyle \frac{d^{k-1}}{d{t}^{k-1}}}f\left(t\right)\right\}\left(\eta \right)\hfill \\ \hfill & =\left(i(\eta -m-ncot\beta )sin\beta +cos\beta {\displaystyle \frac{d}{d\eta }}\right){\left(i(\eta -m-ncot\beta )sin\beta +cos\beta {\displaystyle \frac{d}{d\eta }}\right)}^{k-1}{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)\hfill \\ \hfill & ={\left(i(\eta -m-ncot\beta )sin\beta +cos\beta {\displaystyle \frac{d}{d\eta }}\right)}^k{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right),\hfill \end{array}$$

which finishes the proof. □

We collect the observations regarding Theorem 2 in the remark:

Remark 1.

• For $m=n=0$ and $\beta ={\displaystyle \frac{\pi }{2}}$, we get

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\mathbb{O}}^{({\displaystyle \frac{\pi }{2}},0,0)}\left\{{\displaystyle \frac{d^r}{d{t}^r}}f\left(t\right)\right\}\left(\eta \right)& ={\left(i\eta \right)}^r{\mathcal{F}}_{\mathbb{O}}^{({\displaystyle \frac{\pi }{2}},0,0)}\left\{f\right\}\left(\eta \right)\hfill \\ \hfill & ={\left(i\eta \right)}^r\mathcal{F}\left\{f\right\}\left(\eta \right).\hfill \end{array}$$

(25)

Equation (25) coincides with the FT of the nth derivative.

• Selecting $m=n=0$ results in

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\mathbb{O}}^{(\beta ,0,0)}\left\{{\displaystyle \frac{d^r}{d{t}^r}}f\left(t\right)\right\}\left(\eta \right)& ={\left(i\eta sin\beta +cos\beta {\displaystyle \frac{d}{d\eta }}\right)}^r{\mathcal{F}}_{\mathbb{O}}^{(\beta ,0,0)}\left\{f\right\}\left(\eta \right)\hfill \\ \hfill & ={\left(i\eta sin\beta +cos\beta {\displaystyle \frac{d}{d\eta }}\right)}^r{\mathcal{F}}_{\beta }\left\{f\right\}\left(\eta \right),\hfill \end{array}$$

(26)

which is the FrFT of the nth derivative.

Next, we obtain the result.

Theorem 3.

Let ${\mathcal{K}}_{(\beta ,m,n)}$ be the kernel of the OFrFT. We denote ${\overline{D}}^r$ by the following:

$$\begin{array}{c}\hfill {\overline{D}}^r={\left({\displaystyle \frac{\partial }{\partial t}}-i(tcot\beta +mcsc\beta )\right)}^r,\end{array}$$

(27)

then for $\forall r\in \mathbb{N}$, we have

(i) 

${\overline{D}}^r{\mathcal{K}}_{(\beta ,m,n)}={(-i\eta csc\beta )}^r{\mathcal{K}}_{(\beta ,m,n)}$.

(ii) 

${\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{{\overline{D^{\ast }}}^{r}f\left(t\right)\right\}\left(\eta \right)={(-i\eta csc\beta )}^r{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right).$

Here, ${\overline{D}}^{\ast }=-({\displaystyle \frac{\partial }{\partial t}}+i(tcot\beta +mcsc\beta ))$.

Proof. 

(i) Direct computations yield that

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial t}}\left({\mathcal{K}}_{(\beta ,m,n)}\right)& ={\displaystyle \frac{\partial }{\partial t}}\left(A_{\beta }{e}^{{\displaystyle \frac{i}{2}}(({\eta }^2+t^2+m^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta ))}\right)\hfill \\ \hfill & =(itcot\beta +imcsc\beta -i\eta csc\beta ){\mathcal{K}}_{(\beta ,m,n)}.\hfill \end{array}$$

(28)

Consequently,

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{\partial }{\partial t}}-i(tcot\beta +mcsc\beta )\right){\mathcal{K}}_{(\beta ,m,n)}(t,\eta )& =(-i\eta csc\beta ){\mathcal{K}}_{(\beta ,m,n)}.\hfill \end{array}$$

(29)

By continuing in this way, we arrive at

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{\partial }{\partial t}}-i(tcot\beta +mcsc\beta )\right)}^r{\mathcal{K}}_{(\beta ,m,n)}& ={(-i\eta csc\beta )}^r{\mathcal{K}}_{(\beta ,m,n)}.\hfill \end{array}$$

(30)

(ii) Observe that

$$\begin{array}{cc}\hfill & {\int }_{\mathbb{R}}\overline{D}{\mathcal{K}}_{(\beta ,m,n)}(t,\eta )f\left(t\right)dt\hfill \\ \hfill & ={\int }_{\mathbb{R}}\left({\displaystyle \frac{d}{dt}}-i(tcot\beta +mcsc\beta )\right){\mathcal{K}}_{(\beta ,m,n)}(t,\eta )f\left(t\right)dt\hfill \\ \hfill & ={\int }_{\mathbb{R}}{\displaystyle \frac{\partial }{\partial t}}{\mathcal{K}}_{(\beta ,m,n)}(t,\eta )f\left(t\right)dt-{\int }_{\mathbb{R}}i(tcot\beta +mcsc\beta ){\mathcal{K}}_{(\beta ,m,n)}(t,\eta )f\left(t\right)dt\hfill \\ \hfill & =f\left(t\right){\mathcal{K}}_{(\beta ,m,n)}(t,\eta ){{\displaystyle |}}_{-\infty }^{\infty }-{\int }_{\mathbb{R}}{\mathcal{K}}_{(\beta ,m,n)}(t,\eta ){\displaystyle \frac{\partial }{\partial t}}f\left(t\right)dt-{\int }_{\mathbb{R}}i(tcot\beta +mcsc\beta ){\mathcal{K}}_{(\beta ,m,n)}(t,\eta )f\left(t\right)dt\hfill \\ \hfill & =-{\int }_{\mathbb{R}}{\mathcal{K}}_{(\beta ,m,n)}(t,\eta )\left({\displaystyle \frac{\partial }{\partial t}}+i(tcot\beta +mcsc\beta )\right)f\left(t\right)dt\hfill \\ \hfill & ={\int }_{\mathbb{R}}{\mathcal{K}}_{(\beta ,m,n)}(t,\eta )\overline{D^{\ast }}f\left(t\right)dt.\hfill \end{array}$$

(31)

By continuing in this way, we infer that

$$\begin{array}{c}\hfill {\int }_{\mathbb{R}}{\overline{D}}^r{\mathcal{K}}_{(\beta ,m,n)}(t,\eta )f\left(t\right)dt={\int }_{\mathbb{R}}{\mathcal{K}}_{(\beta ,m,n)}(t,\eta ){\overline{D^{\ast }}}^{r}f\left(t\right)dt.\end{array}$$

(32)

(iii) It follows from Equation (14) that

$$\begin{array}{c}\hfill {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{{\overline{D^{\ast }}}^{r}f\left(t\right)\right\}\left(\eta \right)={\int }_{\mathbb{R}}{\overline{D}}^r{\mathcal{K}}_{(\beta ,m,n)}(t,\eta )f\left(t\right)dt.\end{array}$$

(33)

Substituting (30) in Equation (33) results in

$$\begin{array}{cc}\hfill {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{{\overline{D^{\ast }}}^{r}f\left(t\right)\right\}\left(\eta \right)& ={(-i\eta csc\beta )}^r{\int }_{\mathbb{R}}{\mathcal{K}}_{(\beta ,m,n)}(t,\eta )f\left(t\right)dt\hfill \\ \hfill & ={(-i\eta csc\beta )}^r{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right).\hfill \end{array}$$

(34)

Thus, the proof is completed. □

4. Convolution and Product Theorems for OFrFT

In [14], the authors proposed the convolution definition pertaining to the OFrFT of two signals $f,g\in L^2\left(\mathbb{R}\right)$ of the form

$$\begin{array}{c}\hfill \left(f{★}_{M}g\right)\left(t\right)={\int }_{\mathbb{R}}{e}^{-i{\displaystyle \frac{t^2}{2}}cot\beta }(\tilde{f}★\tilde{g})\left(y\right)dy.\end{array}$$

(35)

Here, ★ is the classical convolution and $\tilde{f}\left(t\right)=g\left(t\right)e^{-i{\displaystyle \frac{t^2}{2}}cot\beta }$ and $\tilde{g}\left(t\right)=g\left(t\right)e^{-i{\displaystyle \frac{t^2}{2}}cot\beta }$, respectively. With this definition, they then established the convolution theorem in the form

$$\begin{array}{c}\hfill {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f{★}_{M}g\right\}\left(\eta \right)=\sqrt{{\displaystyle \frac{1-icot\beta }{2\pi }}}{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right){\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{g\right\}\left(\eta \right)e^{-{\displaystyle \frac{i}{2}}({\eta }^{2}cot\beta +2\eta (n-m cot \beta ))}.\end{array}$$

(36)

Due to Equation (36), it is straightforward to see that their convolution involves the multiplication of their OFrFT and a phase factor.

In this part, we establish two types of convolution theorems for the OFrFT, which are quite different from those proposed [14]. We will see that the convolution theorems are a simple multiplication of the OFrFT and the FT. For this aim, we first introduce the following definition:

Definition 6.

The convolution operation of two functions $f,g\in L^2\left(\mathbb{R}\right)$ in the OFrFT domain is defined by

$$\left(f{\ast }_{o}g\right)\left(t\right)={\int }_{\mathbb{R}}f\left(y\right)g(t-y)e^{{\displaystyle \frac{i}{2}}{(t-y)}^{2}cot\beta }{e}^{-{\displaystyle \frac{i}{2}}{t}^{2}cot\beta }{e}^{-iym csc \beta }dy.$$

(37)

With Definition 6, this leads to the following theorem:

Theorem 4.

Suppose that $f,g\in L^2\left(\mathbb{R}\right)$, then the OFrFT of the convolution of f and g is given by

$$\begin{array}{c}\hfill {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f{\ast }_{o}g\right\}\left(\eta \right)={\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{g\right\}\left(\eta \right)\mathcal{F}\left\{f\right\}(\eta csc\beta ).\end{array}$$

(38)

Proof. 

From Equation (14), it follows that

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f{\ast }_{o}g\right\}\left(\eta \right)\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}\left(f{\ast }_{o}g\right)\left(t\right)e^{{\displaystyle \frac{i}{2}}(({\eta }^2+t^2+m^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta ))}dt\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}f\left(y\right)g(t-y)e^{{\displaystyle \frac{i}{2}}{(t-y)}^{2}cot\beta }{e}^{-{\displaystyle \frac{i}{2}}{t}^{2}cot\beta }{e}^{-iym csc \beta }\hfill \\ \hfill & \times e^{{\displaystyle \frac{i}{2}}(({\eta }^2+t^2+m^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta ))}dydt.\hfill \end{array}$$

(39)

Substituting $z=t-y$ into Equation (39) results in

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f{\ast }_{o}g\right\}\left(\eta \right)\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}f\left(y\right)g\left(z\right)e^{{\displaystyle \frac{i}{2}}{z}^{2}cot\beta }{e}^{-iym csc \beta }{e}^{{\displaystyle \frac{i}{2}}(({\eta }^2+m^2)cot\beta +2(z+y)(m-\eta ) csc \beta +2\eta (n-m cot \beta ))}dydz\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}f\left(y\right)g\left(z\right)e^{{\displaystyle \frac{i}{2}}(({\eta }^2+z^2+m^2)cot\beta +2z(m-\eta ) csc \beta +2\eta (n-m cot \beta ))}{e}^{-iy\eta csc \beta }dydz\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}g\left(z\right)e^{{\displaystyle \frac{i}{2}}(({\eta }^2+z^2+m^2)cot\beta +2z(m-\eta ) csc \beta +2\eta (n-m cot \beta ))}dz{\int }_{\mathbb{R}}f\left(y\right)e^{-iy\eta csc \beta }dy\hfill \\ \hfill & ={\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{g\right\}\left(\eta \right)\mathcal{F}\left\{f\right\}(\eta csc\beta ),\hfill \end{array}$$

(40)

and the proof is complete. □

Remark 2.

In Theorem 4, it can easily be seen that the proposed convolution theorem is the product of the OFrFT and the FT. This form is closer to the convolution theorem for the FT.

The alternative structure of the convolution operation and its theorem is demonstrated as below.

Definition 7.

Let two functions $f,g\in L^2\left(\mathbb{R}\right)$. The convolution of f and g in the OFrFT domain is defined by

$$\begin{array}{c}\hfill \left(f{\ast }_{o}g\right)\left(t\right)=e^{-{\displaystyle \frac{i}{2}}(t^{2}cot\beta +2tm csc \beta )}(\tilde{f}\ast g)\left(t\right).\end{array}$$

(41)

Here, $\tilde{f}\left(t\right)=f\left(t\right)e^{{\displaystyle \frac{i}{2}}(t^{2}cot\beta +2tm csc \beta )}$ and * is the classical convolution operation.

This definition gives the next result.

Theorem 5.

For any functions $f,g\in L^1\left(\mathbb{R}\right)$, then

$$\begin{array}{c}\hfill {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f{\ast }_{o}g\right\}\left(\eta \right)={\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)\mathcal{F}\left\{g\right\}(\eta csc\beta ).\end{array}$$

(42)

Proof. 

From the definition of OFrFT described by Equation (14), we see that

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f{\ast }_{o}g\right\}\left(\eta \right)\hfill \\ \hfill & ={\int }_{\mathbb{R}}\left(f{\ast }_{o}g\right)\left(t\right)A_{\beta }{e}^{{\displaystyle \frac{i}{2}}(({\eta }^2+t^2+m^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta ))}dt\hfill \\ \hfill & ={\int }_{\mathbb{R}}(\tilde{f}\ast g)\left(t\right)A_{\beta }{e}^{{\displaystyle \frac{i}{2}}(({\eta }^2+m^2)cot\beta -2t\eta csc \beta +2\eta (n-m cot \beta ))}dt\hfill \\ \hfill & ={\int }_{\mathbb{R}}{A}_{\beta }{e}^{{\displaystyle \frac{i}{2}}(({\eta }^2+m^2)cot\beta +2\eta (n-m cot \beta ))}(\tilde{f}\ast g)\left(t\right)e^{-it\eta csc \beta }dt\hfill \\ \hfill & =A_{\beta }{e}^{{\displaystyle \frac{i}{2}}(({\eta }^2+m^2)cot\beta +2\eta (n-m cot \beta ))}\mathcal{F}\left\{\tilde{f}\ast g\right\}(\eta csc\beta )\hfill \\ \hfill & =A_{\beta }{e}^{{\displaystyle \frac{i}{2}}(({\eta }^2+m^2)cot\beta +2\eta (n-m cot \beta ))}\mathcal{F}{\displaystyle \{}f\left(t\right)e^{{\displaystyle \frac{i}{2}}(t^{2}cot\beta +2tm csc \beta )}\ast g{\displaystyle \}}(\eta csc\beta ).\hfill \end{array}$$

(43)

Applying the convolution theorem for the FT and Equation (16) results in

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}{\displaystyle \{}f{\ast }_{o}g{\displaystyle \}}\left(\eta \right)\hfill \\ \hfill & =A_{\beta }{e}^{{\displaystyle \frac{i}{2}}(({\eta }^2+m^2)cot\beta +2\eta (n-m cot \beta ))}\mathcal{F}{\displaystyle \{}f\left(t\right)e^{{\displaystyle \frac{i}{2}}(t^{2}cot\beta +2tm csc \beta )}{\displaystyle \}}(\eta csc\beta )\mathcal{F}\left\{g\right\}(\eta csc\beta )\hfill \\ \hfill & ={\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)\mathcal{F}\left\{g\right\}(\eta csc \beta ).\hfill \end{array}$$

The proof is complete. □

Remark 3.

It is easy to see that there is a slight difference between Theorems 4 and 5.

Based on Definition 7, we obtain the product theorem associated with the OFrFT, as expressed below.

Theorem 6.

Let $f,g\in L^1\left(\mathbb{R}\right)$, then one gets

$$\begin{array}{c}\hfill e^{{\displaystyle \frac{i}{2}}{\eta }^{2}cot\beta }\left(\widetilde{{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}}\left\{f\right\}\ast \mathcal{F}\left\{g\right\}\right)\left(\eta \right)={\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\left(t\right)g(tcsc\beta -(n-m cot \beta ))\right\}\left(\eta \right),\end{array}$$

(44)

where

$$\begin{array}{c}\hfill \widetilde{{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}}\left\{f\right\}\left(y\right)={\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\left(t\right)e^{-{\displaystyle \frac{i}{2}}{y}^{2}cot\beta }\right\}\left(y\right).\end{array}$$

Proof. 

The left-hand side of Equation (44) will lead to

$$\begin{array}{cc}\hfill & e^{{\displaystyle \frac{i}{2}}{\eta }^{2}cot\beta }\left(\widetilde{{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}}\left\{f\right\}\ast \mathcal{F}\left\{g\right\}\right)\left(\eta \right)\hfill \\ \hfill & =e^{{\displaystyle \frac{i}{2}}{\eta }^{2}cot\beta }{\int }_{\mathbb{R}}\widetilde{{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}}\left\{f\right\}\left(y\right)\mathcal{F}\left\{g\right\}(\eta -y)dy\hfill \\ \hfill & =e^{{\displaystyle \frac{i}{2}}{\eta }^{2}cot\beta }{A}_{\beta }{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}f\left(t\right)e^{-{\displaystyle \frac{i}{2}}{y}^{2}cot\beta }{e}^{{\displaystyle \frac{i}{2}}\left((y^2+m^2+t^2)cot\beta +2t(m-y)csc\beta +2y(n-m cot \beta )\right)}\mathcal{F}\left\{g\right\}(\eta -y)dtdy\hfill \\ \hfill & =e^{{\displaystyle \frac{i}{2}}{\eta }^{2}cot\beta }{A}_{\beta }{\int }_{\mathbb{R}}f\left(t\right)e^{{\displaystyle \frac{i}{2}}\left((m^2+t^2)cot\beta +2t(m-y)csc\beta +2y(n-m cot \beta )\right)}dt{\int }_{\mathbb{R}}\mathcal{F}\left\{g\right\}(\eta -y)dy.\hfill \end{array}$$

(45)

Equation (45) may be rewritten as

$$\begin{array}{cc}\hfill & e^{{\displaystyle \frac{i}{2}}{\eta }^{2}cot\beta }\left(\widetilde{{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}}\left\{f\right\}\ast \mathcal{F}\left\{g\right\}\right)\left(\eta \right)\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}f\left(t\right)e^{{\displaystyle \frac{i}{2}}(({\eta }^2+t^2+m^2)cot\beta +2t(m-y)csc\beta +2y(n-m cot \beta ))}\mathcal{F}\left\{g\right\}(\eta -y)dtdy.\hfill \end{array}$$

(46)

Substituting $v=\eta -y$ into the above expression results in

$$\begin{array}{cc}\hfill & e^{{\displaystyle \frac{i}{2}}{\eta }^{2}cot\beta }\left(\widetilde{{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}}\left\{f\right\}\ast \mathcal{F}\left\{g\right\}\right)\left(\eta \right)\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}f\left(t\right)e^{{\displaystyle \frac{i}{2}}\left(({\eta }^2+t^2+m^2)cot\beta +2t(m-(\eta -v))csc\beta +2(\eta -v)(n-m cot \beta )\right)}\mathcal{F}\left\{g\right\}\left(v\right)dvdt\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}f\left(t\right)e^{{\displaystyle \frac{i}{2}}\left(({\eta }^2+t^2+m^2)cot\beta +2t(m-\eta +v) csc \beta +2(\eta -v)(n-m cot \beta )\right)}\mathcal{F}\left\{g\right\}\left(v\right)dvdt\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}{\int }_{\mathbb{R}}f\left(t\right)e^{{\displaystyle \frac{i}{2}}\left(({\eta }^2+t^2+m^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta )\right)}\mathcal{F}\left\{g\right\}\left(v\right)e^{i\left(tv csc \beta -v(n-m cot \beta )\right)}dvdt\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)e^{{\displaystyle \frac{i}{2}}\left(({\eta }^2+t^2+m^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta )\right)}\left({\int }_{\mathbb{R}}\mathcal{F}\left\{g\right\}\left(v\right)e^{i(t csc \beta -(n-m cot \beta \left)\right)v}dv\right)dt.\hfill \end{array}$$

Applying Equation (13) produces

$$\begin{array}{cc}\hfill & e^{{\displaystyle \frac{i}{2}}{\eta }^{2}cot\beta }\left(\widetilde{{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}}\left\{f\right\}\ast \mathcal{F}\left\{g\right\}\right)\left(\eta \right)\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)g\left(tcsc\beta -(n-mcot\beta )\right)e^{{\displaystyle \frac{i}{2}}\left(({\eta }^2+t^2+m^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta )\right)}dt\hfill \\ \hfill & ={\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\left(t\right)g(tcsc\beta -(n-mcot\beta \left)\right)\right\}\left(\eta \right),\hfill \end{array}$$

which finishes the proof. □

5. Sampling Formula

The authors of [22,23] discussed the sampling formula for the FrFT and then our work [11] utilized this formula to solve the generalized heat equations. This part will construct a generalization of the sampling formula in the context of the OFrFT. To achieve this, we first recall a class of functions whose OFrFT is zero outside a finite interval $[-\sigma ,\sigma ]$. These functions are often referred to as band-limited $\sigma$. In this case, a positive number $\sigma$ is called a bandwidth of the function. Now, let us begin by introducing the following important definition:

Definition 8.

We call signal $f\left(t\right)$ the band-limited σ for the OFrFT, if there is a positive number σ satisfying ${\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)=0$ for every $\left|\eta \right|>\sigma$.

As an immediate consequence of the above definition, we get the following important result:

Theorem 7.

Suppose that $f\left(t\right)$ is band-limited by σ concerning the OFrFT such that

$$\begin{array}{c}\hfill f\left(t\right)={\int }_{-\sigma }^{\sigma }{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)\overline{{\mathcal{K}}_{(\beta ,m,n)}(t,\eta )}d\eta ,\end{array}$$

(47)

then f has the following expansion:

$$\begin{array}{cc}\hfill & f\left(t\right)\hfill \\ \hfill & =csc\beta e^{{\displaystyle \frac{i}{2}}(t^{2}cot\beta +2tm csc \beta )}\sum _{l\in \mathbb{Z}}{e}^{{\displaystyle \frac{i}{2}}{\displaystyle (}{{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}}^{2}cot\beta +2{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}mcsc\beta {\displaystyle )}}f{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}{\displaystyle \frac{sin{\displaystyle (}t\sigma -l\pi {\displaystyle )}csc\beta }{{\displaystyle (}t\sigma -l\pi {\displaystyle )}csc\beta }}.\hfill \end{array}$$

(48)

Proof. 

According to Equations (14) and (47), we obtain

$$\begin{array}{cc}\hfill f\left(t\right)& ={\int }_{-\sigma }^{\sigma }{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)\overline{{\mathcal{K}}_{(\beta ,m,n)}(t,\eta )}d\eta \hfill \\ \hfill & ={\int }_{-\sigma }^{\sigma }{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)\overline{A_{\beta }}{e}^{-{\displaystyle \frac{i}{2}}\left(({\eta }^2+t^2+m^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta )\right)}d\eta \hfill \\ \hfill & =\overline{A_{\beta }}{e}^{-{\displaystyle \frac{i}{2}}\left((t^2+m^2)cot\beta +2tm csc \beta \right)}{\int }_{-\sigma }^{\sigma }{\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)e^{-{\displaystyle \frac{i}{2}}\left({\eta }^{2}cot\beta -2t\eta csc \beta +2\eta (n-m cot \beta )\right)}d\eta .\hfill \end{array}$$

(49)

On the other hand, we have

$$\begin{array}{c}\hfill \mathcal{F}\left\{f\right\}\left(\eta \right)=\sum _{l\in \mathbb{Z}}{\displaystyle \frac{\sqrt{2\pi }}{2\sigma }}f\left({\displaystyle \frac{l\pi }{\sigma }}\right)e^{-{\displaystyle \frac{il\pi \eta }{\sigma }}}.\end{array}$$

(50)

Setting $\eta$ to $\eta csc\beta$ in Equation (50), it is easily seen that

$$\begin{array}{c}\hfill \mathcal{F}\left\{f\right\}(\eta csc\beta )=\sum _{l\in \mathbb{Z}}{\displaystyle \frac{\sqrt{2\pi }}{2\sigma }}f\left({\displaystyle \frac{l\pi }{\sigma }}\right)e^{-i{\displaystyle \frac{l\pi \eta csc\beta }{\sigma }}}.\end{array}$$

(51)

Relation (51) can be rewritten in the form

$$\begin{array}{cc}\hfill & \mathcal{F}{\displaystyle \{}{e}^{{\displaystyle \frac{i}{2}}(t^{2}cot\beta +2tm csc \beta )}f\left(t\right){\displaystyle \}}(\eta csc\beta )\hfill \\ \hfill & =\sum _{l\in \mathbb{Z}}{\displaystyle \frac{\sqrt{2\pi }}{2\sigma }}{e}^{{\displaystyle \frac{i}{2}}{\displaystyle (}cot\beta {{\displaystyle )}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}}^2+2{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}m csc \beta {\displaystyle )}}f{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}{e}^{-i{\displaystyle \frac{l\pi \eta csc\beta }{\sigma }}}.\hfill \end{array}$$

(52)

Multiplying both sides of Equation (52) by $A_{\beta }{e}^{{\displaystyle \frac{i}{2}}\left(({\eta }^2+m^2)cot\beta +2\eta (n-m cot \beta )\right)}$ produces

$$\begin{array}{cc}\hfill & A_{\beta }{e}^{{\displaystyle \frac{i}{2}}\left(({\eta }^2+m^2)cot\beta +2\eta (n-m cot \beta )\right)}\mathcal{F}{\displaystyle \{}{e}^{{\displaystyle \frac{i}{2}}(t^{2}cot\beta +2tm csc \beta )}f\left(t\right){\displaystyle \}}(\eta csc\beta )\hfill \\ \hfill & =A_{\beta }{e}^{{\displaystyle \frac{i}{2}}\left(({\eta }^2+m^2)cot\beta +2\eta (n-m cot \beta )\right)}\sum _{l\in \mathbb{Z}}{\displaystyle \frac{\sqrt{2\pi }}{2\sigma }}{e}^{{\displaystyle \frac{i}{2}}{\displaystyle (}{{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}}^{2}cot\beta +2{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}m csc \beta {\displaystyle )}}f{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}{e}^{-i{\displaystyle \frac{l\pi \eta csc\beta }{\sigma }}}.\hfill \end{array}$$

(53)

Inserting (16) into the left-hand side of Equation (53), we immediately obtain

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)\hfill \\ \hfill & =A_{\beta }{e}^{{\displaystyle \frac{i}{2}}\left(({\eta }^2+m^2)cot\beta +2\eta (n-m cot \beta )\right)}\sum _{l\in \mathbb{Z}}{\displaystyle \frac{\sqrt{2\pi }}{2\sigma }}{e}^{{\displaystyle \frac{i}{2}}{\displaystyle (}{{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}}^{2}cot\beta +2{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}m csc \beta {\displaystyle )}}f{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}{e}^{-i{\displaystyle \frac{l\pi \eta csc\beta }{\sigma }}}.\hfill \end{array}$$

(54)

Further, substituting (54) into (49), we infer that

$$\begin{array}{cc}\hfill f\left(t\right)& =\overline{A_{\beta }}{e}^{-{\displaystyle \frac{i}{2}}((t^2+m^2) cot \beta +2tm csc \beta )}{\int }_{-\sigma }^{\sigma }{A}_{\beta }{e}^{{\displaystyle \frac{i}{2}}(({\eta }^2+m^2) cot \beta +2\eta (n-m cot \beta ))}\sum _{l\in \mathbb{Z}}{\displaystyle \frac{\sqrt{2\pi }}{2\sigma }}\hfill \\ \hfill & \times e^{{\displaystyle \frac{i}{2}}{\displaystyle (}{{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}}^2 cot \beta +2{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}m csc \beta {\displaystyle )}}f{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}{e}^{-i{\displaystyle \frac{l\pi \eta csc \beta }{\sigma }}}{e}^{-{\displaystyle \frac{i}{2}}{\displaystyle (}{\eta }^2 cot \beta -2t\eta csc \beta +2\eta {\displaystyle (}n-m cot \beta {\displaystyle )}{\displaystyle )}}d\eta \hfill \\ \hfill & ={\displaystyle |}{A}_{\beta }{{\displaystyle |}}^2{e}^{-{\displaystyle \frac{i}{2}}{\displaystyle (}{\displaystyle (}{t}^2+m^2{\displaystyle )} cot \beta +2tm csc \beta {\displaystyle )}}{\int }_{-\sigma }^{\sigma }f{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}\sum _{l\in \mathbb{Z}}{\displaystyle \frac{\sqrt{2\pi }}{2\sigma }}{e}^{{\displaystyle \frac{i}{2}}{\displaystyle (}{{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}}^2 cot \beta +2{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}m csc \beta {\displaystyle )}}\hfill \\ \hfill & \times e^{-i{\displaystyle \frac{l\pi \eta csc \beta }{\sigma }}}{e}^{it\eta csc \beta }d\eta \hfill \\ \hfill & ={\displaystyle |}{A}_{\beta }{{\displaystyle |}}^2\sqrt{2\pi }{e}^{-{\displaystyle \frac{i}{2}}{\displaystyle (}{t}^2 cot \beta +2tm csc \beta {\displaystyle )}}\sum _{l\in \mathbb{Z}}{\displaystyle \frac{\sqrt{2\pi }}{2\sigma }}{e}^{{\displaystyle \frac{i}{2}}{\displaystyle (}{{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}}^2 cot \beta +2{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}m csc \beta {\displaystyle )}}f{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}\hfill \\ \hfill & \times {\displaystyle (}{\displaystyle \frac{1}{2\sigma }}{\int }_{-\sigma }^{\sigma }{e}^{i{\displaystyle (}t-{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}\eta csc \beta }d\eta {\displaystyle )}.\hfill \end{array}$$

(55)

Relation (55) may be expressed as

$$\begin{array}{c}\hfill f\left(t\right)=2\pi {\displaystyle |}{A}_{\beta }{{\displaystyle |}}^2{e}^{{\displaystyle \frac{i}{2}}{\displaystyle (}{t}^2 cot \beta +2tm csc \beta {\displaystyle )}}\sum _{l\in \mathbb{Z}}{e}^{{\displaystyle \frac{i}{2}}{\displaystyle (}{{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}}^2 cot \beta +2{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}m csc \beta {\displaystyle )}}f{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}{\displaystyle \frac{sin{\displaystyle (}t\sigma -l\pi {\displaystyle )} csc \beta }{{\displaystyle (}t\sigma -l\pi {\displaystyle )} csc \beta }},\end{array}$$

(56)

which finishes the proof. □

Remark 4.

• By setting $m=n=0$ in Equation (48), we obtain

$$\begin{array}{c}\hfill f\left(t\right)=csc\beta e^{{\displaystyle \frac{i}{2}}(t^{2}cot\beta )}\sum _{l\in \mathbb{Z}}{e}^{{\displaystyle \frac{i}{2}}{\displaystyle (}{{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}}^{2}cot\beta {\displaystyle )}}f{\displaystyle (}{\displaystyle \frac{l\pi }{\sigma }}{\displaystyle )}{\displaystyle \frac{sin{\displaystyle (}t\sigma -l\pi {\displaystyle )}csc\beta }{{\displaystyle (}t\sigma -l\pi {\displaystyle )}csc\beta }},\end{array}$$

(57)

which is known as the sampling formula for the FrFT.

• For $m=n=0$ and $\beta ={\displaystyle \frac{\pi }{2}}$, we get

$$\begin{array}{c}\hfill f\left(t\right)=\sum _{l\in \mathbb{Z}}f\left({\displaystyle \frac{l\pi }{\sigma }}\right){\displaystyle \frac{sin(t\sigma -l\pi )}{(t\sigma -l\pi )}}.\end{array}$$

(58)

which is known as the Shannon–Whittaker sampling theorem [27].

The next result is the following:

Theorem 8.

Under the assumption as in Theorem 7, we have

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)\hfill \\ \hfill & ={\displaystyle \frac{\pi }{\sigma }}{A}_{\beta }{e}^{{\displaystyle \frac{i}{2}}(({\eta }^2+m^2)cot\beta +2\eta (n-m cot \beta ))}\sum _{l\in \mathbb{Z}}{e}^{{\displaystyle \frac{i}{2}}\left({\left({\displaystyle \frac{l\pi }{\sigma }}\right)}^{2}cot\beta +2\left({\displaystyle \frac{l\pi }{\sigma }}\right)m csc \beta \right)}f\left({\displaystyle \frac{l\pi }{\sigma }}\right)\hfill \\ \hfill & \times e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{{(m-\eta )}^2{csc}^2\beta }{{cot}^2\beta }}}erf\left({\displaystyle \frac{\sigma csc\beta }{2\sqrt{-icot\beta }}}\right)cos\left(\sigma {\displaystyle \frac{(m-\eta ){csc}^2\beta }{cot\beta }}-l\pi csc\beta \right).\hfill \end{array}$$

(59)

Proof. 

Including relation (48) into relation (14), it follows that

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}f\left(t\right)e^{{\displaystyle \frac{i}{2}}(({\eta }^2+t^2+m^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta ))}dt\hfill \\ \hfill & =A_{\beta }{\int }_{\mathbb{R}}2\pi {\left|A_{\beta }\right|}^2{e}^{{\displaystyle \frac{i}{2}}(t^2 cot \beta +2tm csc \beta )}\sum _{l\in \mathbb{Z}}{e}^{{\displaystyle \frac{i}{2}}\left({\left({\displaystyle \frac{l\pi }{\sigma }}\right)}^2 cot \beta +2\left({\displaystyle \frac{l\pi }{\sigma }}\right)m csc \beta \right)}\hfill \\ \hfill & \times f\left({\displaystyle \frac{l\pi }{\sigma }}\right){\displaystyle \frac{sin(t\sigma -l\pi )csc\beta }{(t\sigma -l\pi )csc\beta }}{e}^{{\displaystyle \frac{i}{2}}(({\eta }^2+t^2+m^2)cot\beta +2t(m-\eta ) csc \beta +2\eta (n-m cot \beta ))}dt\hfill \\ \hfill & =2\pi A_{\beta }{\left|A_{\beta }\right|}^2{e}^{{\displaystyle \frac{i}{2}}(({\eta }^2+m^2)cot\beta +2\eta (n-m cot \beta ))}\sum _{l\in \mathbb{Z}}{e}^{{\displaystyle \frac{i}{2}}\left({\left({\displaystyle \frac{l\pi }{\sigma }}\right)}^{2}cot\beta +2\left({\displaystyle \frac{l\pi }{\sigma }}\right)m csc \beta \right)}f\left({\displaystyle \frac{l\pi }{\sigma }}\right)\hfill \\ \hfill & \times {\int }_{\mathbb{R}}{\displaystyle \frac{sin(t\sigma -l\pi )csc\beta }{(t\sigma -l\pi )csc\beta }}{e}^{{\displaystyle \frac{i}{2}}(t^{2}cot\beta +2tm csc \beta )}{e}^{-it\eta csc\beta }dt.\hfill \end{array}$$

(60)

On the other hand, we have

$$\begin{array}{cc}\hfill & {\int }_{\mathbb{R}}{\displaystyle \frac{sin(t\sigma -l\pi )csc\beta }{(t\sigma -l\pi )csc\beta }}{e}^{{\displaystyle \frac{i}{2}}(t^{2}cot\beta +2tm csc \beta )-it\eta csc \beta }dt\hfill \\ \hfill & ={\int }_{\mathbb{R}}{\displaystyle \frac{sin(t\sigma -l\pi )csc\beta }{(t\sigma -l\pi )csc\beta }}{e}^{{\displaystyle \frac{i}{2}}cot\beta t^2+itm csc \beta -it\eta csc \beta }dt\hfill \\ \hfill & ={\int }_{\mathbb{R}}{\displaystyle \frac{sin(t\sigma -l\pi ) csc \beta }{(t\sigma -l\pi ) csc \beta }}{e}^{{\displaystyle \frac{i}{2}} cot \beta (t^2+2{\displaystyle \frac{(m-\eta )csc\beta t}{cot\beta }})}dt\hfill \\ \hfill & ={\int }_{\mathbb{R}}{\displaystyle \frac{sin(t\sigma -l\pi )csc\beta }{(t\sigma -l\pi )csc\beta }}{e}^{{\displaystyle \frac{i}{2}}cot\beta ({(t-{\displaystyle \frac{(m-\eta )csc\beta }{cot\beta }})}^2-{\displaystyle \frac{{(m-\eta )}^2{csc}^2\beta }{{cot}^2\beta }})}dt\hfill \\ \hfill & =e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{{(m-\eta )}^2{csc}^2\beta }{{cot}^2\beta }}}{\int }_{\mathbb{R}}{\displaystyle \frac{sin(t\sigma -l\pi )csc\beta }{(t\sigma -l\pi )csc\beta }}{e}^{{\displaystyle \frac{i}{2}}cot\beta {(t-{\displaystyle \frac{(m-\eta )csc\beta }{cot\beta }})}^2}dt.\hfill \end{array}$$

(61)

Let $v=t-{\displaystyle \frac{(m-\eta )csc\beta }{cot\beta }}$, the above expression becomes

$$\begin{array}{cc}\hfill & {\int }_{\mathbb{R}}{\displaystyle \frac{sin(t\sigma -l\pi )csc\beta }{(t\sigma -l\pi )csc\beta }}{e}^{{\displaystyle \frac{i}{2}}(t^{2}cot\beta +2tm csc \beta )-it\eta csc \beta }dt\hfill \\ \hfill & =e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{{(m-\eta )}^2{csc}^2\beta }{{cot}^2\beta }}}{\int }_{\mathbb{R}}{\displaystyle \frac{sin(\sigma v+\sigma \frac{(m-\eta ) csc \beta }{cot\beta }-l\pi )csc\beta }{(\sigma v+\sigma \frac{(m-\eta ) csc \beta }{cot\beta }-l\pi )csc\beta }}{e}^{{\displaystyle \frac{i}{2}}cot\beta v^2}dv.\hfill \end{array}$$

(62)

Further, we have

$$\begin{array}{cc}\hfill & {\int }_{\mathbb{R}}{\displaystyle \frac{sin(t\sigma -l\pi )csc\beta }{(t\sigma -l\pi )csc\beta }}{e}^{{\displaystyle \frac{i}{2}}(t^{2}cot\beta +2tm csc \beta )-it\eta csc\beta }dt\hfill \\ \hfill & =e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{{(m-\eta )}^2{csc}^2\beta }{{cot}^2\beta }}}{\int }_{\mathbb{R}}{\displaystyle \frac{sin(\sigma csc\beta v+\sigma \frac{(m-\eta ){csc}^2\beta }{cot\beta }-l\pi csc\beta )}{(\sigma csc\beta v+\sigma \frac{(m-\eta ){csc}^2\beta }{cot\beta }-l\pi csc\beta )}}{e}^{{\displaystyle \frac{i}{2}}cot\beta v^2}dv\hfill \\ \hfill & =e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{{(m-\eta )}^2{csc}^2\beta }{{cot}^2\beta }}}{\displaystyle \frac{\pi }{\sigma csc\beta }}erf\left({\displaystyle \frac{\sigma csc\beta }{2\sqrt{-icot\beta }}}\right)cos\left(\sigma {\displaystyle \frac{(m-\eta ){ csc}^{ 2}\beta }{cot\beta }}-l\pi csc\beta \right).\hfill \end{array}$$

(63)

Substituting relation (63) into relation (60) results in

$$\begin{array}{cc}\hfill & {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)\hfill \\ \hfill & =2\pi A_{\beta }{\left|A_{\beta }\right|}^2{e}^{{\displaystyle \frac{i}{2}}(({\eta }^2+m^2)cot\beta +2\eta (n-m cot \beta ))}\sum _{l\in \mathbb{Z}}{e}^{{\displaystyle \frac{i}{2}}\left({\left({\displaystyle \frac{l\pi }{\sigma }}\right)}^{2}cot\beta +2\left({\displaystyle \frac{l\pi }{\sigma }}\right)mcsc\beta \right)}f\left({\displaystyle \frac{l\pi }{\sigma }}\right)\hfill \\ \hfill & \times e^{-{\displaystyle \frac{i}{2}}{\displaystyle \frac{{(m-\eta )}^2{csc}^2\beta }{{cot}^2\beta }}}{\displaystyle \frac{\pi }{\sigma csc\beta }}erf\left({\displaystyle \frac{\sigma csc\beta }{2\sqrt{-icot\beta }}}\right)cos\left(\sigma {\displaystyle \frac{(m-\eta ){csc}^2\beta }{cot\beta }}-l\pi csc\beta \right).\hfill \end{array}$$

(64)

Thus, the proof is complete. □

Let us now illustrate the utility of Theorem 7 above by providing the following example:

Example 1.

Consider a function defined by

$$\begin{array}{c}\hfill {\mathcal{F}}_{\mathbb{O}}^{(\beta ,m,n)}\left\{f\right\}\left(\eta \right)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\left|\eta \right|\le 1\hfill \\ 0,\hfill & \mathrm{if}\left|\eta \right|>1.\hfill \end{array}\right.\end{array}$$

(65)

In fact, we have

$$\begin{array}{cc}\hfill f\left(t\right)& =\overline{A_{\beta }}{\int }_{-1}^1{e}^{-{\displaystyle \frac{i}{2}}(({\eta }^2+t^2+m^2)cot\beta +2t(m-\eta )csc\beta +2\eta (n-m cot \beta ))}d\eta \hfill \\ \hfill & =\overline{A_{\beta }}{e}^{-{\displaystyle \frac{i}{2}}((t^2+m^2)cot\beta +2tmcsc\beta )}{\int }_{-1}^1{e}^{-{\displaystyle \frac{i}{2}}cot\beta \left({\eta }^2-2({\displaystyle \frac{tcsc\beta }{cot\beta }}-{\displaystyle \frac{(n-mcot\beta )}{cot\beta }})\eta \right)}d\eta .\hfill \end{array}$$

(66)

This equation may be expressed as

$$\begin{array}{cc}\hfill & f\left(t\right)\hfill \\ \hfill & =\overline{A_{\beta }}{e}^{-{\displaystyle \frac{i}{2}}((t^2+m^2)cot\beta +2tm csc \beta )}{\int }_{-1}^1{e}^{-{\displaystyle \frac{i}{2}}cot\beta {\left(\eta -({\displaystyle \frac{tcsc\beta }{cot\beta }}-{\displaystyle \frac{(n-m cot \beta )}{cot\beta }})\right)}^2-{\left({\displaystyle \frac{t csc \beta }{cot\beta }}-{\displaystyle \frac{(n-mcot\beta )}{cot\beta }}\right)}^2)}d\eta \hfill \\ \hfill & =\overline{A_{\beta }}{e}^{-{\displaystyle \frac{i}{2}}((t^2+m^2)cot\beta +2tm csc \beta -{\displaystyle \frac{{(t csc \beta -(n-mcot\beta ))}^2}{cot\beta }})}{\int }_{-1}^1{e}^{-{\displaystyle \frac{i}{2}}cot\beta {\left(\eta -({\displaystyle \frac{tcsc\beta }{cot\beta }}-{\displaystyle \frac{(n-mcot\beta )}{cot\beta }})\right)}^2}d\eta \hfill \\ \hfill & =2\overline{A_{\beta }}{e}^{-{\displaystyle \frac{i}{2}}((t^2+m^2)cot\beta +2tm csc \beta -{\displaystyle \frac{{(t csc \beta -(n-mcot\beta ))}^2}{cot\beta }})}{\int }_0^1{e}^{-{\displaystyle \frac{i}{2}}cot\beta {\left(\eta -({\displaystyle \frac{tcsc\beta }{cot\beta }}-{\displaystyle \frac{(n-mcot\beta )}{cot\beta }})\right)}^2}d\eta .\hfill \end{array}$$

(67)

Let

$$\begin{array}{c}\hfill u=\eta -\left({\displaystyle \frac{tcsc\beta }{cot\beta }}-{\displaystyle \frac{(n-m cot \beta )}{cot\beta }}\right),\end{array}$$

(68)

Thus, Equation (67) changes to

$$\begin{array}{cc}\hfill & f\left(t\right)\hfill \\ \hfill & =2\overline{A_{\beta }}{e}^{-{\displaystyle \frac{i}{2}}((t^2+m^2)cot\beta +2tm csc \beta -{\displaystyle \frac{{(tcsc\beta -(n-m cot \beta ))}^2}{cot\beta }})}{\int }_{-({\displaystyle \frac{tcsc\beta }{cot\beta }}-{\displaystyle \frac{(n-m cot \beta )}{cot\beta }})}^{1-({\displaystyle \frac{tcsc\beta }{cot\beta }}-{\displaystyle \frac{(n-m cot \beta )}{cot\beta }})}{e}^{-{\displaystyle \frac{i}{2}}cot\beta u^2}du\hfill \\ \hfill & =2\overline{A_{\beta }}{e}^{-{\displaystyle \frac{i}{2}}((t^2+m^2)cot\beta +2tm csc \beta -{\displaystyle \frac{{(tcsc\beta -(n-mcot\beta ))}^2}{cot\beta }})}\hfill \\ \hfill & \times {\displaystyle \frac{\sqrt{\pi }}{\sqrt{\frac{icot\beta }{2}}}}\left(erf\left(1-{\displaystyle \frac{tcsc\beta }{cot\beta }}+{\displaystyle \frac{(n-m cot \beta )}{cot\beta }}\right)+erf\left({\displaystyle \frac{tcsc\beta }{cot\beta }}-{\displaystyle \frac{(n-m cot \beta )}{cot\beta }}\right)\right)\hfill \\ \hfill & =2\sqrt{1+itan\beta }{e}^{-{\displaystyle \frac{i}{2}}((t^2+m^2)cot\beta +2tmcsc\beta -{\displaystyle \frac{{(tcsc\beta -(n-mcot\beta ))}^2}{cot\beta }})}\hfill \\ \hfill & \times \left(erf\left(1-{\displaystyle \frac{t}{cos\beta }}+ntan\beta -m\right)+erf\left({\displaystyle \frac{t}{cos\beta }}-(ntan\beta -m)\right)\right).\hfill \end{array}$$

(69)

Here, $erf\left(z\right)={\int }_0^z{e}^{-t^2}dt$.

Now, substituting Equation (69) into Equation (48), we infer that

$$\begin{array}{cc}\hfill & f\left(t\right)\hfill \\ \hfill & =\sqrt{1+itan\beta }{e}^{{\displaystyle \frac{i}{2}}(t^{2}cot\beta +2tm csc \beta )}\sum _{l\in \mathbb{Z}}{e}^{{\displaystyle \frac{i}{2}}\left({\left({\displaystyle \frac{l\pi }{\sigma }}\right)}^{2}cot\beta +2\left({\displaystyle \frac{l\pi }{\sigma }}\right)m csc \beta \right)}{\displaystyle \frac{sin(t\sigma -l\pi )csc\beta }{(t\sigma -l\pi )csc\beta }}\hfill \\ \hfill & \times e^{-{\displaystyle \frac{i}{2}}({\left({\displaystyle \frac{l\pi }{\sigma }}\right)}^2+m^2)cot\beta +{\displaystyle \frac{2l\pi m}{\sigma }}csc\beta -{\displaystyle \frac{{(\frac{l\pi csc\beta }{\sigma }-(n-mcot\beta ))}^2}{cot\beta }})}\hfill \\ \hfill & \times \left(erf\left(1-{\displaystyle \frac{l\pi }{\sigma cos\beta }}+ntan\beta -m\right)+erf\left({\displaystyle \frac{l\pi }{\sigma cos\beta }}-(ntan\beta -m)\right)\right).\hfill \end{array}$$

(70)

Figure 1 and Figure 2 demonstrate the comparison of $f\left(t\right)$ in Equation (69) using the partial series with sampling Formula (70).

6. Conclusions

In this work, we introduced the OFrFT and investigated its fundamental properties, which have not appeared in the existing literature to date. The convolution and product theorems related to this transform were constructed. A sampling formula involving the OFrFT was established. Lastly, we verified the sampling formula for the OFrFT by providing a simple example.