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A new structure of an integral operator associated with trigonometric Dunkl settings

Abstract

In this paper, we discuss a generalization to the Cherednik–Opdam integral operator to an abstract space of Boehmians. We introduce sets of Boehmians and establish delta sequences and certain class of convolution products. Then we prove that the extended Cherednik–Opdam integral operator is linear, bijective and continuous with respect to the convergence of the generalized spaces of Boehmians. Moreover, we derive embeddings and discuss properties of the generalized theory. Moreover, we obtain an inversion formula and provide several results.

Introduction and preliminaries

Generalized functions were designed by Sobolev and Schwartz to fulfil the apparent requirements of science. Being an extension to the concept of ordinary functions, the theory of generalized functions gives rise to many fruitful results in partial differential equations, yet at the same time they are arbitrarily singular. In literature, the notion of generalized functions have witnessed a volcanic growth in PDEs, physics, engineering, mathematical physics, theoretical stochastic analysis and some numerical aspects as well. Typically generalized functions are defined as continuous linear mappings on appropriately defined spaces of test functions, nevertheless, Boehmians are introduced as quotients of convolution products similar to the concept of field of quotients (see, e.g., [15] and [610]). Although the construction of a Boehmian space might be obtained by convolution products and delta sequences of shrinking supports to the origin, it may not be possible to define a notion of a Boehmian when the delta sequences fail to vanish on an open set. The idea of such a construction has led to many important ideas on the support of a Boehmian and the abelian-type theorems of the integral transform operators.

Here and throughout, it being understood conventionally that \(\mathbb{C}\), \(\mathbb{R}\) and \(\mathbb{N}\) are the sets of complex numbers, real numbers and positive integers, respectively. For arbitrary but fixed real parameters α and β subject to the constraints \(\alpha \geq \beta \geq -1/2, \alpha >-1/2\) and \(\lambda \in \mathbb{C}\), the Opdam–Cherednik theory is a theory based on the Opdam–Cherednik normalized eigen function \(G_{\lambda }^{ ( \alpha,\beta ) }\) of the Dunkl–Cherednik differential–difference operator \(F^{ ( \alpha,\beta ) }\), where

$$\begin{aligned} F^{ ( \alpha,\beta ) }G_{\lambda }^{ ( \alpha, \beta ) } ( t ) =i\lambda G_{\lambda }^{ ( \alpha,\beta ) } ( t ), \end{aligned}$$

provided \(G_{\lambda }^{ ( \alpha,\beta ) } ( 0 ) =1, F^{ ( \alpha,\beta ) }f ( t ) =f^{{\prime }} ( t ) + \{ ( \alpha -\beta ) \coth t+ ( 2\beta +1 ) \coth 2t \} F ( t ) -pf ( -t )\), \(p=\alpha +\beta +1\) and \(F ( t ) =f ( t ) -f ( -t ) \) (see, e.g., [11]). An efficient relation between the hypergeometric and the Jacobi functions, \(G_{\lambda }^{ ( \alpha,\beta ) } \) and \(\varphi _{\lambda }^{ ( \alpha,\beta ) }\), was established as

$$\begin{aligned} G_{\lambda }^{ ( \alpha,\beta ) } ( t ) = \varphi _{\lambda }^{ ( \alpha,\beta ) } ( t ) -\frac{1}{p-i\lambda }\frac{\partial }{\partial t}\varphi _{\lambda }^{ ( \alpha,\beta ) } ( t ) , \end{aligned}$$

where \(\varphi _{\lambda }^{ ( \alpha,\beta ) } ( t ) ={}_{2}F_{1} ( \frac{p+i\lambda }{2},\frac{p-i\lambda }{2};\alpha +1;- \sinh ^{2}t ) \), \(_{2}F_{1}\) being the hypergeometric function. A translation formula, in this division, was introduced as

$$\begin{aligned} \tau _{x}^{ ( \alpha,\beta ) }f ( y ) = \int _{\mathbb{R}}f ( w ) \,d\mu _{x,y}^{ ( \alpha, \beta ) } ( w ), \end{aligned}$$
(1)

where

$$\begin{aligned} &d\mu _{x,y}^{ ( \alpha,\beta ) } ( w ) = \textstyle\begin{cases} K_{\alpha,\beta } ( x,y,w ) A_{\alpha,\beta } ( w ) \,dw, & xy\neq 0, \\ d\delta _{x} ( w ), & y=0, \\ d\delta _{y} ( w ), & x=0,\end{cases}\displaystyle \\ &K_{\alpha,\beta } ( x,y,w ) =A_{xyw} \int _{0}^{-2 \alpha }g_{x,y,w,\theta }^{\alpha -\beta -1} \bigl( 1-\sigma _{x,y,w}^{ \theta }+\sigma _{x,w,y}^{\theta }+ \sigma _{w,y,x}^{\theta }+h_{x,y,w, \theta }^{\beta,\alpha } \bigr) \sin ^{2\beta }\theta \,d\theta, \\ & A_{xyw}= \frac{\mu _{\alpha +\beta }}{ \vert \sinh x\sinh y\sinh w \vert ^{2\alpha }},\qquad h_{x,y,w, \theta }^{\beta,p}= \frac{\alpha \coth x\coth y\coth w\sin ^{2}\theta }{\beta +\frac{1}{2}},\\ &\quad x,y,w\in \mathbb{R}\backslash \{ 0 \} , \quad\text{and} \\ &g_{x,y,w,\theta } =1-\cosh ^{2}x-\cosh ^{2}y\cosh ^{2}w+2\cosh x\cosh y \cosh w\cos \theta, \end{aligned}$$
(2)

satisfy the triangular inequality \(\vert \vert x \vert - \vert y \vert \vert < \vert w \vert < \vert x \vert + \vert y \vert \), provided that

$$\begin{aligned} \sigma _{x,y,w}^{\theta }=\textstyle\begin{cases} \frac{\cosh x+\cosh y-\cosh w\cos \theta }{\sinh w\sinh y}, & xy\neq 0, \\ 0, & xy=0,\end{cases}\displaystyle \end{aligned}$$

for \(x,y,w\in \mathbb{R},\theta \in [ 0,1 ] \) and \(K_{\alpha,\beta } ( x,y,w ) =0\) otherwise. It will be very useful to report here that a change of variables in \(K_{\alpha,\beta }\), for \(x,y,w\in \mathbb{R}\), yields (see, e.g., [12])

$$\begin{aligned} K_{\alpha,\beta } ( x,y,w ) =K_{\alpha,\beta } ( y,x,w ) =K_{\alpha,\beta } ( -x,w,y ) =K_{\alpha,\beta } ( -w,y,-x ). \end{aligned}$$

In the literature, the Opdam–Cherednik integral operator is defined, on the space \(C_{c} (\mathbb{R} ) \) of continuous functions with compact support on \(\mathbb{R}\), as a Fourier integral operator in trigonometric Dunkl settings given by [13]

$$\begin{aligned} \gamma ( f ) ( \lambda ) = \int _{\mathbb{R}}f ( t ) G_{\lambda }^{ ( \alpha,\beta ) } ( -t ) \vec{d}_{\alpha,\beta }t, \end{aligned}$$
(3)

where \(\alpha \geq \beta \geq -\frac{1}{2}\), \(\alpha > -\frac{1}{2}\) and \(\lambda \in \mathbb{C} \). The inversion formula of the Opdam–Cherednik integral operator can be recovered from the Opdam–Cherednik integral operator as

$$\begin{aligned} \gamma ^{-1} ( h ) ( \lambda ) = \int _{\mathbb{R}}h ( \lambda ) G_{\lambda }^{ ( \alpha,\beta ) } ( t ) \biggl( 1- \frac{p}{i\lambda } \biggr) \frac{ 1 }{8\pi \vert C_{\alpha,\beta } ( \lambda ) \vert ^{2}}\,d \lambda. \end{aligned}$$

The Roe and the Paley–Wiener theorems in the context of Cherednik operators were established by using tempered distributions with spectral gaps in the Opdam–Cherednik operator (see, e.g., [14, 15]). On the other hand, uncertainty principles and local uncertainty principles of Donoho–Strak type were derived by Achak and Daher [16]. Further, in an attractive perspective, they have adequately established certain analogs of Hardy, Beurling, Cowling-Price, Gelfand–Shilov and Miyachi theorems, with the aid of composition properties of the Opdam–Cherednik operator. However, various investigations and real applications of this integral operator may be observed in [11, 12, 1720] and the references therein.

Although there were discussed various integral operators on different spaces of Boehmians, the theory of the Cherednik–Opdam integral operator of a Boehmian has not yet been reported in the literature. The starting point in such an approach relies on a convolution theorem which, in addition to delta sequences, allows embeddings to act as isomorphisms between the classical spaces \(L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) and \(L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ), \vec{d}_{\alpha,\beta }x=A_{\alpha,\beta } ( \vert x \vert )\,dx, A_{\alpha,\beta } ( \vert x \vert ) =\sinh ^{2\alpha +1} \vert x \vert \cosh ^{2\beta +1} \vert x \vert \), and the generalized spaces \(\beta _{1}^{ ( \alpha,\beta ) }\) and \(\beta _{2}^{ ( \alpha,\beta ) }\) of Boehmians, respectively. Here and hereafter, we will be concerned with themes in the context of Boehmian spaces and the framework of the Cherednik–Opdam integral operator on the real line. We will consider a Cherednik–Opdam set of delta sequences and provide various axioms to legitimate the Cherednik–Opdam sets of Boehmians. Similar argument is also applied to the Cherednik–Opdam inversion formula. To be more precise, we expand our results into three sections. In Sect. 1, we recall the general description of the Opdam hypergeometric functions, convolution products and some related results. In Sect. 2, we define delta sequences and prove numerous results that presumably exhibits the Cherednik–Opdam spaces of Boehmians. In Sect. 3, we discuss the generalized theory of the Cherednik–Opdam integral and its inversion formula in the class of Boehmians.

Opdam–Cherednik generalized spaces

In this section, we generate two imperative sets of Boehmians and extend the Opdam–Cherednik integral operator to the given sets. For certain deterministic needs, we introduce Opdam–Cherednik sets of delta sequences, convolution products, convolution theorems and establish vital axioms for such extension. By \(L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \), we denote the set of all measurable functions such that the integral formula

$$\begin{aligned} \Vert f \Vert _{\alpha,\beta }= \int _{\mathbb{R}} \bigl\vert f ( x ) \bigr\vert \vec{d}_{\alpha,\beta }x \end{aligned}$$
(4)

is finite. Likewise, by \(D ( \mathbb{R} ) \) we denote the subspace of the measurable space \(L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) of test functions of compact supports over \(\mathbb{R}\). The convolution product between two suitable functions \(f_{1}\) and \(f_{2}\) in the space \(L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) is defined, when the integral exists, as [13]

$$\begin{aligned} f_{1}\ast _{\alpha,\beta }f_{2} ( x ) = \int _{\mathbb{R}}\tau _{x}^{ ( \alpha,\beta ) }f_{1} ( -y ) f_{2} ( y )\vec{d}_{\alpha,\beta }y. \end{aligned}$$
(5)

In the course of the following two lemmas, we recite some properties of the product \(\ast _{\alpha,\beta }\) as follows.

Lemma 1

([13, Remark 4.8])

Let \(f_{1},f_{2},f_{3}\in L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \). Then the following holds:

$$\begin{aligned} f_{1}\ast _{\alpha,\beta }f_{2}=f_{2}\ast _{\alpha,\beta }f_{1}\quad \textit{and}\quad ( f_{1}\ast _{\alpha,\beta }f_{2} ) \ast _{ \alpha,\beta }f_{3}=f_{1} \ast _{\alpha,\beta } ( f_{2}\ast _{ \alpha,\beta }f_{3} ). \end{aligned}$$

For \(p=q=r=1\), the following result is very beneficial for the sequel.

Lemma 2

([13, Proposition 4.10])

Let \(f_{1},f_{2}\in L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \). Then \(f_{1}\ast _{\alpha,\beta }f_{2}\in L^{1} ( \mathbb{R},\vec{d}_{ \alpha,\beta }x ) \) and

$$\begin{aligned} \Vert f_{1}\ast _{\alpha,\beta }f_{2} \Vert _{\alpha, \beta }\leq C_{\alpha,\beta } \Vert f_{1} \Vert _{\alpha, \beta } \Vert f_{2} \Vert _{\alpha,\beta }, \end{aligned}$$

where the coefficients are given by

$$\begin{aligned} C_{\alpha,\beta }=\textstyle\begin{cases} 4+ \frac{\Gamma ( \alpha +1 ) \Gamma ( \beta +\frac{1}{2} ) }{\Gamma ( \alpha +\frac{1}{2} ) \Gamma ( \beta +1 ) }, & \alpha >\beta >-\frac{1}{2}, \\ \frac{5}{2}, & \alpha =\beta >-\frac{1}{2}. \end{cases}\displaystyle \end{aligned}$$
(6)

As delta sequences are essential parts in this treatment, the Opdam–Cherednik set of delta sequences can be presented as follows.

Definition 3

By \(\Delta ^{ ( \alpha,\beta ) }\), we denote the subset of \(D ( \mathbb{R} ) \) consisting of all sequences \(( \delta _{n} ) \) such that the identities \(P_{1}, P_{2}\) and \(P_{3} \) hold:

$$\begin{aligned} &P_{1}: \int _{\mathbb{R}}\delta _{n} ( x ) G_{0}^{ ( \alpha,\beta ) } ( -x ) \vec{d}_{\alpha, \beta }x=1\quad\text{for every }n\in \mathbb{N}, \\ &P_{2}: \Vert \delta _{n} \Vert _{\alpha,\beta }\leq M \quad\text{for some constant }M>0\text{ and every }n\in \mathbb{N}, \\ &P_{3}:\lim_{n\rightarrow \infty } \int _{ \vert x \vert >\epsilon } \bigl\vert \delta _{n} ( x ) \bigr\vert \vec{d}_{\alpha,\beta }x=0 \quad\text{for every real number }\epsilon >0. \end{aligned}$$

We have the following assertion.

Proposition 4

The set \(\Delta ^{ ( \alpha,\beta ) }\) is an Opdam–Cherednik set of delta sequences.

Proof

To show that \(\Delta ^{ ( \alpha,\beta ) }\) is an Opdam–Cherednik set of delta sequences we have to show that \(P_{1} \)\(P_{3}\) hold for all \(\Delta ^{ ( \alpha,\beta ) }\) elements. Let \(( \delta _{n} ), ( \theta _{n} ) \in \Delta ^{ ( \alpha,\beta ) }\). Then, by the convolution theorem \(\gamma ( f\ast _{\alpha,\beta }g ) ( \lambda ) =\gamma ( f ) ( \lambda ) \gamma (g ) ( \lambda ) \) which can deduced from [13, Proposition 4.9] and [21], we write

$$\begin{aligned} \gamma ( \delta _{n}\ast _{\alpha,\beta }\theta _{n} ) ( 0 ) =\gamma ( \delta _{n} ) ( 0 ) \gamma ( \theta _{n} ) ( 0 ). \end{aligned}$$
(7)

Therefore, we rewrite (7) in an explicit form as

$$\begin{aligned} \int _{\mathbb{R}} ( \delta _{n}\ast _{\alpha,\beta } \theta _{n} ) ( x ) G_{0}^{\alpha,\beta } ( -x ) \vec{d}_{\alpha,\beta }x = \int _{R}\delta _{n} ( x ) G_{0}^{\alpha,\beta } ( -x ) \vec{d}_{ \alpha,\beta }x \int _{\mathbb{R}}\theta _{n} ( x ) G_{0}^{\alpha,\beta } ( -x ) \vec{d}_{\alpha, \beta }x. \end{aligned}$$

Hence, from the previous equation it can be easily inferred that

$$\begin{aligned} \int _{\mathbb{R}} ( \delta _{n}\ast _{\alpha,\beta } \theta _{n} ) ( x ) G_{0}^{\alpha,\beta } ( -x ) \vec{d}_{\alpha,\beta }x=1. \end{aligned}$$
(8)

Hence, \(P_{1}\) is completely proved. To prove \(P_{2}\), let \(( \delta _{n} ) \) and \(( \theta _{n} ) \) be in \(\Delta ^{ ( \alpha,\beta ) }\). Then we have \(\Vert \delta _{n} \Vert _{\alpha,\beta }\leq M_{1}\) and \(\Vert \theta _{n} \Vert _{\alpha,\beta }\leq M_{2}\) for some real numbers \(M_{1}\) and \(M_{2}\). Therefore, from Lemma 2, we get \(\Vert \delta _{n}\ast _{\alpha,\beta }\theta _{n} \Vert _{\alpha,\beta }\leq M\) where \(M=C_{\alpha,\beta }M_{1}M_{2}\). Finally, if \(\lim_{n\rightarrow \infty }\int _{ \vert x \vert >\epsilon } \vert \delta _{n} ( x ) \vert \vec{d}_{\alpha,\beta }x=0\) and \(\lim_{n\rightarrow \infty }\int _{ \vert x \vert >\epsilon } \vert \theta _{n} ( x ) \vert \vec{d}_{\alpha,\beta }x=0\), then we have

$$\begin{aligned} \lim_{n\rightarrow \infty } \int _{ \vert x \vert >\epsilon } \bigl\vert ( \delta _{n}\ast _{\alpha,\beta } \theta _{n} ) ( x ) \bigr\vert \vec{d}_{\alpha, \beta }x \leq \lim_{n\rightarrow \infty } \int _{ \vert x \vert >\epsilon } \bigl\vert \delta _{n} ( x ) \bigr\vert \vec{d}_{\alpha,\beta }x \lim_{n\rightarrow \infty } \int _{ \vert x \vert >\epsilon } \bigl\vert \theta _{n} ( x ) \bigr\vert \vec{d}_{\alpha, \beta }x. \end{aligned}$$

Hence, we have obtained

$$\begin{aligned} \lim_{n\rightarrow \infty } \int _{ \vert x \vert >\epsilon } \bigl\vert ( \delta _{n}\ast _{\alpha,\beta } \theta _{n} ) ( x ) \bigr\vert \vec{d}_{\alpha, \beta }x=0. \end{aligned}$$

This completes the proof of the proposition. □

We now establish the prerequisite axioms for generating the Boehmian space \(\beta _{1}^{ ( \alpha,\beta ) }\) with the set \(L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \), the subset \(D ( \mathbb{R} ) \), the product \(\ast _{\alpha,\beta }\) and the set \(\Delta ^{ ( \alpha,\beta ) }\) of delta sequences.

Theorem 5

Let \(f,g,h,f_{n}\rightarrow f\) as \(n\rightarrow \infty \) in \(L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \), \(\delta,\theta \in D ( \mathbb{R} ) \) and \(\alpha \in \mathbb{C}\). Then we have:

  1. (i)

    \(\alpha ( f\ast _{\alpha,\beta }g ) = ( \alpha f ) \ast _{\alpha,\beta }g\).

  2. (ii)

    \(( f+g ) \ast _{\alpha,\beta }\delta =f \ast _{\alpha,\beta }\delta +g\ast _{\alpha,\beta }\delta \).

  3. (iii)

    \(f\ast _{\alpha,\beta }g=g\ast _{\alpha,\beta }f\), and \(( f\ast _{\alpha,\beta }g ) \ast _{\alpha,\beta }h=f \ast _{\alpha,\beta } ( g\ast _{\alpha,\beta }h ) \).

  4. (iv)

    \(f_{n}\ast _{\alpha,\beta }\delta \rightarrow f \ast _{\alpha,\beta }\delta \) as \(n\rightarrow \infty \).

Proof

The proofs of (i) and (ii) are straightforward. The proof of (iii) is consistent with the proof of Lemma 2. Hence, the proofs are omitted. The proof of (iv) follows from Lemma 2 and the fact that

$$\begin{aligned} \bigl\Vert ( f_{n}-f ) \ast _{\alpha,\beta }\delta \bigr\Vert _{\alpha,\beta }\leq C_{\alpha,\beta } \Vert f_{n}-f \Vert _{\alpha,\beta } \Vert \delta \Vert _{ \alpha,\beta }\rightarrow 0\quad\text{as }n\rightarrow \infty. \end{aligned}$$

This finishes the proof of the theorem. □

Finally, we establish the following lemma.

Lemma 6

Let \(f\in L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) and \(( \delta _{n} ) \in \Delta ^{ ( \alpha,\beta ) }\). Then \(f\ast _{\alpha,\beta }\delta _{n}\rightarrow f\) as \(n\rightarrow \infty \) in \(L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \).

Proof

Let \(( \delta _{n} ) \in \Delta ^{ ( \alpha,\beta ) }\) and \(f\in L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) be given. Then by [11, Lemma 4.6] we have

$$\begin{aligned} \bigl\Vert \tau _{x}^{ ( \alpha,\beta ) }f \bigr\Vert _{ \alpha,\beta } \leq C_{\alpha,\beta } \Vert f \Vert _{ \alpha,\beta }, \end{aligned}$$
(9)

where \(C_{\alpha,\beta }\) has a significance of (6). Also, from [12] we see that the function \(G_{0}^{ ( \alpha,\beta ) }\) is strictly positive and bounded above and, for every \(\lambda \in \mathbb{R}\), we have

$$\begin{aligned} \bigl\vert G_{\lambda }^{ ( \alpha,\beta ) } ( x ) \bigr\vert \leq G_{0} ( x ) \quad\text{for every }x \in \mathbb{R}. \end{aligned}$$
(10)

Therefore, by Definition 3 and Fubini’s theorem, we write

$$\begin{aligned} &\Vert f\ast _{\alpha,\beta }\delta _{n}-f \Vert _{ \alpha,\beta } \\ &\quad\leq \int _{\mathbb{R}} \biggl( \int _{\mathbb{R}} \bigl\vert \bigl( \tau _{x}^{ ( \alpha, \beta ) }f ( -y ) -f ( x ) G_{0}^{ ( \alpha,\beta ) } ( -y ) \bigr) \delta _{n} ( y ) \vec{d}_{\alpha,\beta }y \bigr\vert \biggr) \vec{d}_{ \alpha,\beta }x \\ &\quad\leq \int _{\mathbb{R}} \biggl( \int _{\mathbb{R}} \bigl\vert \tau _{x}^{ ( \alpha,\beta ) }f ( -y ) -f ( x ) G_{0}^{ ( \alpha,\beta ) } ( -y ) \bigr\vert \bigl\vert \delta _{n} ( y ) \bigr\vert \vert \vec{d}_{\alpha,\beta }y \vert \biggr) \vec{d}_{\alpha,\beta }x \\ &\quad\leq \int _{\mathbb{R}} \biggl( \int _{\mathbb{R}} \bigl( \bigl\vert \tau _{x}^{ ( \alpha,\beta ) }f ( -y ) \bigr\vert + \bigl\vert f ( x ) G_{0}^{ ( \alpha,\beta ) } ( -y ) \bigr\vert \bigr) \bigl\vert \delta _{n} ( y ) \bigr\vert \vert \vec{d}_{\alpha,\beta }x \vert \biggr) \vec{d}_{\alpha, \beta }x. \end{aligned}$$

Let C be an upper bound for \(G_{0}^{ ( \alpha,\beta ) }\), then, by [15, Proposition 4.4], we have

$$\begin{aligned} &\Vert f\ast _{\alpha,\beta }\delta _{n}-f \Vert _{ \alpha,\beta } \\ &\quad\leq \int _{\mathbb{R}} \biggl( \int _{\mathbb{R}} \bigl( \bigl\vert \tau _{y}^{ ( \alpha, \beta ) }f ( -x ) \bigr\vert +C \bigl\vert f ( x ) \bigr\vert \bigr) \bigl\vert \delta _{n} ( y ) \bigr\vert \vert \vec{d}_{\alpha,\beta }y \vert \biggr) \vec{d}_{\alpha,\beta }x \\ &\quad\leq \int _{\mathbb{R}} \bigl( C_{\alpha,\beta } \Vert f \Vert _{\alpha,\beta }+C \Vert f \Vert _{ \alpha,\beta } \bigr) \bigl\vert \delta _{n} ( y ) \bigr\vert \vert \vec{d}_{\alpha,\beta }y \vert \\ &\quad\leq \bigl( C_{\alpha,\beta } \Vert f \Vert _{\alpha, \beta }+C \Vert f \Vert _{\alpha,\beta } \bigr) \int _{\mathbb{R}} \bigl\vert \delta _{n} ( y ) \bigr\vert \vert \vec{d}_{\alpha,\beta }y \vert \\ &\quad\leq \bigl( C_{\alpha,\beta } \Vert f \Vert _{\alpha, \beta }+C \Vert f \Vert _{\alpha,\beta } \bigr) \Vert \delta _{n} \Vert _{\alpha,\beta }\rightarrow 0 \quad\text{as }n\rightarrow \infty. \end{aligned}$$

This finishes the proof of the theorem. □

If \(( \varphi _{n} ) \in L^{1} ( \mathbb{R},\vec{d}_{ \alpha,\beta }x ) \) and \(( \delta _{n} ) \in \Delta ^{ ( \alpha,\beta ) }\), then the pair \(( \varphi _{n},\delta _{n} ) \) \((\text{or }\frac{\varphi _{n}}{\delta _{n}} ) \) is said to be a quotient of sequences if \(\varphi _{n}\ast _{\alpha,\beta }\delta _{m}=\varphi _{m}\ast _{ \alpha,\beta }\delta _{n},\forall n,m\in \mathbb{N}\). Therefore, if \(\frac{\varphi _{n}}{\delta _{n}}\) and \(\frac{g_{n}}{\varepsilon _{n}}\) are quotients of sequences and \(\psi \in L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \), then it is easy to see that

$$\begin{aligned} \frac{\psi \ast _{\alpha,\beta }\delta _{n}}{\delta _{n}} \quad\text{and}\quad \frac{\varphi _{n}\ast _{\alpha,\beta }\delta _{n}+g_{n}\ast _{\alpha,\beta }\delta _{n}}{\delta _{n}\ast _{\alpha,\beta }\varepsilon _{n}} \end{aligned}$$

are quotients of sequences. Further, we can easily check the following equivalence relations:

$$\begin{aligned} \frac{\varphi _{n}}{\delta _{n}\ast _{\alpha,\beta }\psi }\sim \frac{\varphi _{n}\ast _{\alpha,\beta }\psi }{\delta _{n}} \quad\text{and}\quad \frac{\varphi _{n}}{\delta _{n}\ast _{\alpha,\beta }g_{n}}\sim \frac{\varphi _{n}\ast _{\alpha,\beta }g_{n}}{\delta _{n}}. \end{aligned}$$

Two quotients of sequences \(\frac{\varphi _{n}}{\delta _{n}}\) and \(\frac{g_{n}}{\varepsilon _{n}}\) are said to be equivalent if \(\varphi _{n}\ast _{\alpha,\beta }\varepsilon _{m}=g_{m}\ast _{ \alpha,\beta }\delta _{n},\forall n,m\in \mathbb{N}\). The equivalent class \(\breve{w}= ( \frac{\varphi _{n}}{\delta _{n}} ) \) of quotients of sequences containing \(\frac{\varphi _{n}}{\delta _{n}}\) is said to be a Boehmian. The space of such Boehmians is denoted by \(\beta _{1}^{ ( \alpha,\beta ) }\). For two Boehmians \(\breve{w}= ( \frac{\varphi _{n}}{\delta _{n}} ) \) and \(\breve{z}= ( \frac{g_{n}}{\varepsilon _{n}} ) \) in \(\beta _{1}^{ ( \alpha,\beta ) }\), the following are well-defined on \(\beta _{1}^{ ( \alpha,\beta ) }\):

$$\begin{aligned} &\mathrm{(i)} \quad\breve{w}+\breve{z}= \biggl( \frac{\varphi _{n}\ast _{\alpha,\beta }\delta _{n}+g_{n}\ast _{\alpha,\beta }\delta _{n}}{\delta _{n}\ast _{\alpha,\beta }\varepsilon _{n}} \biggr),\qquad \mathrm{(ii)} \quad\beta \breve{w}= \biggl( \frac{\beta \varphi _{n}}{\delta _{n}} \biggr), \\ &\mathrm{(iii)}\quad \breve{w}\ast _{\alpha,\beta }\breve{z}= \biggl( \frac{\varphi _{n}\ast _{\alpha,\beta }g_{n}}{\delta _{n}\ast _{\alpha,\beta }\varepsilon _{n}} \biggr), \qquad \mathrm{(iv)} \quad D^{k}\breve{w}= \biggl( \frac{D^{k}\varphi _{n}}{\delta _{n}} \biggr), \quad\text{and} \\ & \mathrm{(v)}\quad \breve{w}\ast _{\alpha,\beta }\psi = \biggl( \frac{\varphi _{n}\ast _{\alpha,\beta }\psi }{\delta _{n}} \biggr), \end{aligned}$$

where \(k\in \mathbb{R}\), \(\beta \in \mathbb{C}\) and \(D^{k}\breve{w}\) is the kth derivative of and \(\psi \in L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \).

Definition 7

For \(n=1,2,3,\ldots \) and \(\breve{w}_{n},\breve{w}\in \beta _{1}^{ ( \alpha,\beta ) }\), the sequence \(( \breve{w}_{n} ) \) is said to be δ-convergent to , denoted by \(\delta -\lim_{n\rightarrow \infty }\breve{w}_{n}=\breve{w}\), provided there can be found a delta sequence \(( \delta _{n} ) \)such that

$$\begin{aligned} &\mathrm{(a)}\quad ( \breve{w}_{n}\ast _{\alpha,\beta }\delta _{k} ), ( \breve{w}\ast _{\alpha,\beta }\delta _{k} ) \quad\text{in }L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ),\text{ for all }n,k\in \mathbb{N,} \\ &\mathrm{(b)}\quad \lim_{n\rightarrow \infty }\breve{w}_{n}\ast _{ \alpha,\beta } \delta _{k}=\breve{w}\ast _{\alpha,\beta }\delta _{k}\quad \text{in }L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) ,\text{ for every }k\in \mathbb{N}. \end{aligned}$$

Definition 8

For \(n=1,2,3,\ldots \) and \(\breve{w}_{n},\breve{w}\in \beta _{1}^{ ( \alpha,\beta ) }\), the sequence \(( \breve{w}_{n} ) \) is said to be \(\Delta ^{ ( \alpha,\beta ) }\)-convergent to , denoted by \(\Delta ^{\alpha,\beta }\)-\(\lim_{n\rightarrow \infty }\breve{w}_{n}=\breve{w}\), provided there can be found a delta sequence \(( \delta _{n} ) \) such that

$$\begin{aligned} &\mathrm{(a)}\quad ( \breve{w}_{n}-\breve{w} ) \ast _{ \alpha,\beta }\delta _{n}\in L^{1} ( \mathbb{R},\vec{d}_{ \alpha,\beta }x ) \quad( \forall n\in \mathbb{N} ), \\ &\mathrm{(b)}\quad \lim_{n\rightarrow \infty } ( \breve{w}_{n}- \breve{w} ) \ast _{\alpha,\beta }\delta _{n}=0 \quad\text{in }L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ). \end{aligned}$$

Remark 9

Let \(\psi \in L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) and \(( \delta _{n} ) \in \Delta ^{ ( \alpha,\beta ) }\) be fixed. Then we have the mapping

$$\begin{aligned} \psi \rightarrow \breve{w}, \end{aligned}$$
(11)

where \(\breve{w}= ( \frac{\psi \ast _{\alpha,\beta }\delta _{n}}{\delta _{n}} ) \) is an injective mapping from \(L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) into \(\beta _{1}^{ ( \alpha,\beta ) }\).

Therefore, it can be easily checked that \(L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) may be identified as a subspace of \(\beta _{1}^{ ( \alpha,\beta ) }\).

Remark 10

Let \(( \delta _{n} ) \in \Delta ^{ ( \alpha,\beta ) }\). Then, if \(\psi _{n}\rightarrow \psi \) in \(L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) as \(n\rightarrow \infty \), then, for all \(k\in \mathbb{N}\),

$$\begin{aligned} \psi _{n}\ast _{\alpha,\beta }\delta _{k}\rightarrow \psi \ast _{ \alpha,\beta }\delta _{k} \end{aligned}$$

as \(n\rightarrow \infty \). That is, \(\breve{w}_{n}\rightarrow \breve{w}\) in \(\beta _{1}^{ ( \alpha,\beta ) }\) as \(n\rightarrow \infty \).

The above remark states the following.

Theorem 11

The mappings \(\psi \rightarrow \breve{w}\), \(\breve{w}= ( \frac{\psi \ast _{\alpha,\beta }\delta _{n}}{\delta _{n}} ) \), is a continuous embedding of the space \(L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) into the space \(\beta _{1}^{ ( \alpha,\beta ) }\).

Now, let \(L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \), \(D_{E} ( \mathbb{R} ) \) and \(\Delta _{E}^{ ( \alpha,\beta ) }\) be the Opdam–Cherednic operators of the spaces \(L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ),D ( \mathbb{R} ) \) and \(\Delta ^{ ( \alpha,\beta ) }\), respectively. Then we have the following definition.

Definition 12

Let \(F\in L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) and \(\Psi \in D_{E} ( \mathbb{R} ) \). Then between F and Ψ, we define an operation \(\ast _{\alpha,\beta }^{E}\) as

$$\begin{aligned} F\ast _{\alpha,\beta }^{E}\Psi ( x ) = ( F\Psi ) ( x ), \end{aligned}$$
(12)

where \(( F\Psi ) ( x ) =F ( x ) \Psi ( x ) \) is the usual product of two functions.

Accordingly, the construction of the Boehmian space \(\beta _{2}^{ ( \alpha,\beta ) }\) with the sets \(L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \), \(D_{E} ( \mathbb{R} ) \) and \(\Delta _{E}^{ ( \alpha,\beta ) }\) follows from the following theorem.

Theorem 13

Let \(F,G,F_{n},F\) be in \(L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ),\psi _{1}, \psi _{2}\in D_{E} ( \mathbb{R} ), ( \theta _{n}^{E} ) \), \(( \delta _{n}^{E} ) \in \Delta _{E}^{ ( \alpha, \beta ) }\), \(\bar{\alpha }\in \mathbb{C}\) and \(F_{n}\rightarrow F\) as \(n\rightarrow \infty \). Then we have

  1. (i)

    \(\bar{\alpha } ( F\ast _{\alpha,\beta }^{E}G ) = ( \bar{\alpha }F ) \ast _{\alpha,\beta }^{E}G\).

  2. (ii)

    \(( \theta _{n}^{E}\ast _{\alpha,\beta }^{E} \delta _{n}^{E} ) \in \Delta _{E}^{ ( \alpha,\beta ) }\).

  3. (iii)

    \(F\ast _{\alpha,\beta }^{E} ( \psi _{1} \ast _{\alpha,\beta }^{E}\psi _{2} ) = ( F\ast _{\alpha, \beta }^{E}\psi _{1} ) \ast _{\alpha,\beta }^{E}\psi _{2}\).

  4. (iv)

    \(F_{n}\ast _{\alpha,\beta }^{E}\psi _{1} \rightarrow F\ast _{\alpha,\beta }^{E}\psi _{1}\) as \(n\rightarrow \infty \).

  5. (v)

    \(F\ast _{\alpha,\beta }^{E}\delta _{n}^{E} \rightarrow F\) as \(n\rightarrow \infty \) in \(L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \).

  6. (vi)

    \(( F+G ) \ast _{\alpha,\beta }^{E} \psi _{1}=F\ast _{\alpha,\beta }^{E}+G\ast _{\alpha,\beta }^{E} \psi _{1}\).

Proof of (i)

By using (12), two times, we get \(\bar{\alpha } ( F\ast _{\alpha,\beta }^{E}G ) ( \lambda ) =\bar{\alpha } ( FG ) ( \lambda ) = ( \bar{\alpha }F ( \lambda ) ) G ( \lambda ) = ( \bar{\alpha }F ) \ast _{ \alpha,\beta }^{E}G\). This proves (i). To prove (ii), let \(( \theta _{n} ), ( \delta _{n} ) \in \Delta ^{ ( \alpha,\beta ) }\) be such that \(\theta _{n}^{E}=\gamma \theta _{n}\) and \(\delta _{n}^{E}=\gamma \delta _{n}\) for all \(n\in \mathbb{N}\). Then, by [13, Proposition 4.9], we have

$$\begin{aligned} \bigl( \theta _{n}^{E}\ast _{\alpha,\beta }^{E} \delta _{n}^{E} \bigr) ( \lambda ) = \bigl( \gamma \theta _{n}\ast _{ \alpha,\beta }^{E}\gamma \delta _{n} \bigr) ( \lambda ) =\gamma ( \theta _{n}\ast _{\alpha,\beta }\delta _{n} ) ( \lambda ) \in \Delta _{E}^{ ( \alpha,\beta ) } \end{aligned}$$

as \(\Delta ^{ ( \alpha,\beta ) }\) is a delta sequence and \(\theta _{n}\ast _{\alpha,\beta }\delta _{n}\in \Delta ^{ ( \alpha,\beta ) }\). The proofs of (iii)–(vi) follow from [13, Proposition 4.9] and the fact that \(\beta _{1}^{ ( \alpha,\beta ) }\) is a Boehmian space. Hence we omit the details.

The proof of this theorem is therefore finished. □

For every \(( F_{n} ) \in L_{E}^{1} ( \mathbb{R},\vec{d}_{ \alpha,\beta }x ) \) and \(( \delta _{n}^{E} ) \in \Delta _{E}^{ ( \alpha, \beta ) }\), the pair of sequences \(( F_{n},\delta _{n}^{E} ) \) \(( \text{or }\frac{F_{n}}{\delta _{n}^{E}} ) \) is said to be a quotient of sequences in \(\beta _{2}^{ ( \alpha,\beta ) }\) if \(F_{n}\ast _{\alpha,\beta }^{E}\delta _{m}^{E}=F_{m}\ast _{\alpha, \beta }^{E}\delta _{n}^{E} \), for all \(n,m\in \mathbb{N}\). Hence, we may easily check that \(\frac{F\ast _{\alpha,\beta }^{E}\delta _{n}^{E}}{\delta _{n}^{E}}, \frac{F_{n}}{\delta _{n}^{E}\ast _{\alpha,\beta }^{E}F}= \frac{F_{n}\ast _{\alpha,\beta }^{E}F}{\delta _{n}^{E}}, \frac{F_{n}\ast _{\alpha,\beta }^{E}\delta _{n}^{E}+G_{n}\ast _{\alpha,\beta }^{E}\delta _{n}^{E}}{\delta _{n}^{E}\ast _{\alpha,\beta }\theta _{n}^{E}} \frac{F_{n}}{\delta _{n}^{E}\ast _{\alpha,\beta }^{E}G_{n}}\) and \(=\frac{F_{n}\ast _{\alpha,\beta }^{E}G_{n}}{\delta _{n}^{E}}\) when \(\frac{F_{n}}{\delta _{n}^{E}}\) and \(\frac{G_{n}}{\theta _{n}^{E}}\) are quotients of sequences and \(F\in L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \). Moreover, the quotients \(\frac{F_{n}}{\delta _{n}^{E}}\) and \(\frac{G_{n}}{\theta _{n}^{E}}\) are said to be equivalent if \(F_{n}\ast _{\alpha,\beta }^{E}\theta _{m}^{E}=G_{m}\ast _{\alpha, \beta }^{E}\delta _{n}^{E}\) for every \(n,m\in \mathbb{N}\). The equivalent class of quotients of sequences containing \(\frac{F_{n}}{\delta _{n}^{E}}\) is the Boehmian \(\breve{W}= ( \frac{F_{n}}{\delta _{n}^{E}} ) \). The space of such Boehmians is denoted \(\beta _{2}^{ ( \alpha,\beta ) }\). For \(\breve{W}= ( \frac{F_{n}}{\delta _{n}^{E}} ) \) and \(\breve{Z}= ( \frac{G_{n}}{\theta _{n}^{E}} ) \) in \(\beta _{2}^{ ( \alpha,\beta ) }\), addition and scalar multiplication are, respectively, defined as + \(\breve{Z}= ( \frac{F_{n}\ast _{\alpha,\beta }^{E}\delta _{n}^{E}+G_{n}\ast _{\alpha,\beta }^{E}\delta _{n}^{E}}{\delta _{n}^{E}\ast _{\alpha,\beta }^{E}\theta _{n}^{E}} ) \) and \(\beta \breve{W}= ( \frac{\beta F_{n}}{\delta _{n}^{E}} ) \), \(\beta \in \mathbb{C}\). Moreover, we have \(\breve{W}\ast _{\alpha,\beta }^{E}F= ( \frac{F_{n}\ast _{\alpha,\beta }^{E}F}{\delta _{n}^{E}} ) \) for \(F\in L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \).

\(\delta ^{E}\)-convergence: In \(\beta _{2}^{ ( \alpha,\beta ) }\), \(( \breve{W_{n}} ) \) is \(\delta ^{E}\)-convergent to , denoted by \(\delta ^{E}-\lim_{n\rightarrow \infty } \breve{W_{n}}=\breve{W}\), provided there can be found a delta sequence \(( \delta _{n}^{E} ) \) such that

$$\begin{aligned} &\mathrm{( i )} \quad\bigl(\breve{W_{n}}\ast _{\alpha,\beta }^{E} \delta _{k}^{E} \bigr), \bigl( \breve{W}\ast _{\alpha,\beta }^{E} \delta _{k}^{E} \bigr) \text{ in }L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ), \quad\text{for all }n,k\in \mathbb{N,} \\ &\mathrm{( ii )}\quad \lim_{n\rightarrow \infty }\breve{W_{n}}\ast _{ \alpha,\beta }^{E}\delta _{k}=\breve{W}\ast _{\alpha,\beta }\delta _{k} \text{ in }L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ), \quad\text{for every }k\in \mathbb{N}. \end{aligned}$$

\(\Delta _{E}^{ ( \alpha,\beta ) }\)-convergent: In \(\beta _{2}^{ ( \alpha,\beta ) }, (\breve{W_{n}} ) \) is \(\Delta _{E}^{ ( \alpha,\beta ) }\)-convergent to , denoted by \(\Delta _{E}^{ ( \alpha,\beta ) }\)-\(\lim_{n\rightarrow \infty } \breve{W_{n}}=\breve{W}\), provided there can be found a delta sequence \(( \delta _{n}^{E} ) \) such that

$$\begin{aligned} &\mathrm{( i )} \quad(\breve{W_{n}}-\breve{W} ) \ast _{ \alpha,\beta }^{E} \delta _{n}^{E}\in L_{E}^{1} ( \mathbb{R}, \vec{d}_{\alpha,\beta }x ) \quad( \forall n\in \mathbb{N} ), \\ &\mathrm{( ii )}\quad \lim_{n\rightarrow \infty } ( \breve{W_{n}}- \breve{W} ) \ast _{\alpha,\beta }^{E}\delta _{n}^{E}=0\quad \text{in }L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ). \end{aligned}$$

Theorem 14

Let \(( \Omega _{n} ) \in \Delta _{E}^{ ( \alpha,\beta ) }\), \(\Omega _{n}=\gamma \psi _{n}\) for some fixed \(( \psi _{n} ) \in \Delta ^{ ( \alpha,\beta ) }\) and \(W\in L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) where \(W=\gamma \psi,\psi \in L^{1} ( \mathbb{R},\vec{d}_{\alpha, \beta }x )\), then we have the mapping

$$\begin{aligned} W\rightarrow \breve{W}, \end{aligned}$$
(13)

where \(\breve{W}= ( \frac{W\ast _{\alpha,\beta }^{E}\Omega _{n}}{\Omega _{n}} ) \) is an injective mapping from \(L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) into \(\beta _{2}^{ ( \alpha,\beta ) }\).

Hence, from Theorem 14, it can be seen that the space \(L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) may be identified as a subspace of \(\beta _{2}^{ ( \alpha,\beta ) }\). This, indeed, leads to the following results.

Theorem 15

Let \(( \Omega _{n} ) \in \Delta _{E}^{ ( \alpha,\beta ) }\). If \(W_{n}\rightarrow W\) in \(L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) as \(n\rightarrow \infty \), then, for all \(k\in \mathbb{N}\),

$$\begin{aligned} W_{n}\ast _{\alpha,\beta }^{E}\Omega _{k} \rightarrow W\ast _{\alpha, \beta }^{E}\Omega _{k} \end{aligned}$$

as \(n\rightarrow \infty \). That is, \(\breve{W}_{n}\rightarrow \breve{W}\) in \(\beta _{2}^{ ( \alpha,\beta ) }\) as \(n\rightarrow \infty \).

Theorem 16

The mapping \(W\rightarrow \breve{W}\) defined by (13) is a continuous embedding of \(L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) into the space \(\beta _{2}^{ ( \alpha,\beta ) }\).

The generalized Opdam–Cherednik transform

This section discusses a pair of generalized Opdam–Cherednik operators and derive some general properties. Based on the structure of the Boehmian spaces \(\beta _{1}^{ ( \alpha,\beta ) }\) and \(\beta _{2}^{ ( \alpha,\beta ) }\) and the convolution theorem we present the following definition.

Definition 17

The generalized Opdam–Cherednik integral operator of a Boehmian in \(\beta _{1}^{ ( \alpha,\beta ) }\) is the Boehmian in \(\beta _{2}^{ ( \alpha,\beta ) }\) defined by

$$\begin{aligned} \gamma _{E}\breve{w}=\breve{W}, \end{aligned}$$
(14)

where \(\breve{w}= ( \frac{f_{n}}{\delta _{n}} ),f_{n}\in L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ),\delta _{n}\in \Delta ^{ ( \alpha,\beta ) },\forall n\in \mathbb{N}\), and \(\breve{W}= ( \frac{\gamma f_{n}}{\gamma \delta _{n}} ) \).

Theorem 18

Let \(\breve{w}= ( \frac{f_{n}}{\delta _{n}} ) \) be a Boehmian in \(\beta _{1}^{ ( \alpha,\beta ) }\). Then the mapping \(\breve{w}\rightarrow \breve{W}\), defined by \(\breve{W}=\gamma _{E}\breve{w}\), coincides with the corresponding mapping \(\gamma:L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \rightarrow L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \).

Proof

Let \(w\in L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \), then \(\breve{w}= ( \frac{w\ast _{\alpha,\beta }\delta _{n}}{\delta _{n}} ) \) is the identification of w in \(\beta _{1}^{ ( \alpha,\beta ) }\). On the other hand, \(( 14 ) \) and [13, Proposition 4.9] reveal that the Boehmian

$$\begin{aligned} \breve{W}=\gamma _{E}\breve{w}= \biggl( \frac{\gamma ( w\ast _{\alpha,\beta }\delta _{n} ) }{\gamma ( \delta _{n} ) } \biggr) = \biggl( \frac{W\Omega _{n}}{\Omega _{n}} \biggr) = \biggl( \frac{W\ast _{\alpha,\beta }^{E}\Omega _{n}}{\Omega _{n}} \biggr) \end{aligned}$$

can be identified with \(W\in L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) in \(\beta _{1}^{ ( \alpha,\beta ) }\) provided \(\Omega _{n}=\gamma ( \delta _{n} ) \) and \(W=\gamma w\).

The proof is therefore finished. □

We state without proof the following characterization theorem.

Theorem 19

Let \(\breve{w}= ( \frac{f_{n}}{\delta _{n}} ) \) and \(\breve{W}=\gamma _{E}\breve{w}\). Then the mapping \(\breve{w}\rightarrow \breve{W}\) is linear, bijective and continuous with respect to the convergence of the Boehmian spaces. The proof of this theorem is analogous to the proofs given in the literature (see, e.g., [2225]). Hence, it has been omitted.

We introduce the inverse integral operator of the operator \(\gamma _{E}\) as the Boehmian in \(\beta _{1}^{ ( \alpha,\beta ) }\) defined by follows.

Definition 20

Let \(\breve{W}\in \beta _{2}^{ ( \alpha,\beta ) }\), where \(\breve{W}=\gamma _{E}\breve{w},\breve{w}= ( \frac{f_{n}}{\delta _{n}} ), ( f_{n} ) \in L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) and \(( \delta _{n} ) \in \Delta ^{ ( \alpha,\beta ) }\). We define the inverse \(\gamma _{E}\) operator of as

$$\begin{aligned} ( \gamma _{E} ) ^{-1}\breve{W}=\breve{w}. \end{aligned}$$

Theorem 21

The inverse operator \(\breve{w}\rightarrow \breve{W}\) is linear.

Proof

Let and be the Boehmians in \(\beta _{2}^{ ( \alpha,\beta ) }\) such that \(\breve{W}=\gamma _{E}\breve{w}\) and \(\breve{X}=\gamma _{E}\breve{x}\) where \(\breve{w}= ( \frac{\psi _{n}}{\delta _{n}} ),\breve{x}= ( \frac{x_{n}}{\epsilon _{n}} ) \). Then, for all \(n\in \mathbb{N}\) [13, Proposition 4.9], the linearity of the integral leads to

$$\begin{aligned} \breve{W}+\breve{X}= \biggl( \frac{\gamma \psi _{n}\ast _{\alpha,\beta }^{E}\gamma \epsilon _{n}+\gamma x_{n}\ast _{\alpha,\beta }^{E}\gamma \epsilon _{n}}{\gamma \delta _{n}\ast _{\alpha,\beta }^{E}\gamma \epsilon _{n}} \biggr) = \biggl( \frac{\gamma ( \psi _{n}\ast _{\alpha,\beta }\epsilon _{n}+x_{n}\ast _{\alpha,\beta }\delta _{n} ) }{\gamma ( \delta _{n}\ast _{\alpha,\beta }\epsilon _{n} ) } \biggr). \end{aligned}$$

Hence, employing Definition 20 yields

$$\begin{aligned} ( \gamma _{E} ) ^{-1} ( \breve{W}+\breve{X} ) = \biggl( \frac{\psi _{n}\ast _{\alpha,\beta }\epsilon _{n}+x_{n}\ast _{\alpha,\beta }\delta _{n}}{\delta _{n}\ast _{\alpha,\beta }\epsilon _{n}} \biggr) . \end{aligned}$$

Notion of addition in \(\beta _{1}^{ ( \alpha,\beta ) }\) reveals \(( \gamma _{E} ) ^{-1} ( \breve{W}+\breve{X} ) = \breve{w}+\breve{x}\). To complete the proof of the theorem, we, indeed, have to mention that, for some \(\eta \in \mathbb{C}\) and all \(n\in \mathbb{N}\), we have

$$\begin{aligned} ( \gamma _{E} ) ^{-1} ( \eta \breve{W} ) = \eta \breve{w}. \end{aligned}$$

This finishes the proof of the theorem. □

Theorem 22

Let \(\breve{W}\in \beta _{2}^{ ( \alpha,\beta ) },\breve{W}= \gamma _{E}\breve{w},\breve{w}= ( \frac{\psi _{n}}{\epsilon _{n}} ) \) and \(X\in L_{E}^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ),X= \gamma x, x\in L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \). Then we have \(( \gamma _{E} ) ^{-1} ( \breve{W}\ast _{\alpha, \beta }^{E}X ) =\breve{w}\ast _{\alpha,\beta }x\).

Proof

Assume \(\breve{W}\in \beta _{2}^{ ( \alpha,\beta ) }\) where \(\breve{W}=\gamma _{E}\breve{w},\breve{w}= ( \frac{\psi _{n}}{\epsilon _{n}} ) \). Hence, we have

$$\begin{aligned} ( \gamma _{E} ) ^{-1} \bigl( \breve{W}\ast _{\alpha, \beta }^{E}X \bigr) = \biggl( \frac{\gamma \psi _{n}\ast _{\alpha,\beta }^{E}\gamma x}{\digamma \epsilon _{n}} \biggr). \end{aligned}$$

Upon using [13, Proposition 4.9] and Definition 20, we obtain

$$\begin{aligned} ( \gamma _{E} ) ^{-1} \bigl( \breve{W}\ast _{\alpha, \beta }^{E}X \bigr) = ( \gamma _{E} ) ^{-1} \biggl( \frac{\gamma ( \psi _{n}\ast _{\alpha,\beta }x ) }{\gamma \epsilon _{n}} \biggr) = \biggl( \frac{\psi _{n}}{\epsilon _{n}}\ast _{\alpha, \beta }x \biggr) =\breve{w}\ast _{\alpha,\beta }x. \end{aligned}$$

The proof of this theorem is, therefore, finished. □

Theorem 23

Let \(\breve{W}\in \beta _{2}^{ ( \alpha,\beta ) },\breve{W}= \gamma _{E}\breve{w},\breve{w}= ( \frac{\psi _{n}}{\delta _{n}} ) \) and \(X\in L_{E}^{1} (\mathbb{R},\vec{d}_{\alpha,\beta }x ),X= \gamma x,x\in L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \). Then we have \(\gamma _{E} ( \breve{w}\ast _{\alpha,\beta }x ) = \breve{W}\ast _{\alpha,\beta }^{E}X\).

The proof of this theorem is analogous to the proof of Theorem 22. Details are, therefore, omitted.

Theorem 24

Let \(\breve{W}\in \beta _{2}^{ ( \alpha,\beta ) }\), \(\breve{W}=\gamma _{E}\breve{w},\breve{w}= ( \frac{f_{n}}{\delta _{n}} ) \), then is in the range of \(\gamma _{E}\) iff \(\gamma f_{n}\) is in the range of γ, for \((f_{n} )\in L^{1} ( \mathbb{R},\vec{d}_{\alpha, \beta }x ) \) and \((\delta _{n} )\in \Delta ^{ ( \alpha,\beta ) }\).

Proof

Let be in the range of \(\gamma _{E},\breve{W}=\gamma _{E}\breve{w},\breve{w}= ( \frac{f_{n}}{\delta _{n}} ),f_{n}\in L^{1} ( \mathbb{R},\vec{d}_{\alpha,\beta }x ) \) and \(\delta _{n}\in \Delta ^{ ( \alpha,\beta ) }\), for all \(n\in \mathbb{N}\). Then it is clear that \(f_{n}\) is in the range of γ for all \(n\in \mathbb{N}\). On the other hand, if is in the range of \(\gamma _{E}\), then \(\breve{W}=\gamma _{E}\breve{w},\breve{w}= ( \frac{f_{n}}{\delta _{n}} ),f_{n}\in L^{1} ( \mathbb{R}, \vec{d}_{\alpha,\beta }x ),\delta _{n}\in \Delta ^{ ( \alpha,\beta ) },\forall n\in \mathbb{N}\). Therefore, the equivalence relation in \(\beta _{2}^{ ( \alpha,\beta ) }\) leads to

$$\begin{aligned} \gamma f_{n}\ast _{\alpha,\beta }^{E}\gamma \delta _{m}=\gamma f_{m} \ast _{\alpha,\beta }^{E}\gamma \delta _{n}. \end{aligned}$$

Therefore, [13, Proposition 4.9] states that \(\gamma ( f_{n}\ast _{\alpha,\beta }\delta _{m} ) = \gamma ( f_{m}\ast _{\alpha,\beta }\delta _{n} ) \). Hence, it follows that \(f_{n}\ast _{\alpha,\beta }\delta _{m}=f_{m}\ast _{\alpha,\beta } \delta _{n}\). That is, \(\frac{f_{n}}{\delta _{n}}\) is a quotient of sequences and the equivalence class \(( \frac{f_{n}}{\delta _{n}} ) =\breve{w}\) containing \(\frac{f_{n}}{\delta _{n}}\) is the Boehmian in \(\beta _{1}^{ ( \alpha,\beta ) }\) satisfying \(\gamma _{E}\breve{w}=\breve{W}\).

This finishes the proof of the theorem. □

Conclusion

This paper has demonstrated the possibility of extending the Opdam–Cherednik integral operator into a class of generalized functions. It proposes sets of delta sequences and convolution products and investigates the classes of Boehmians. Moreover, this paper considers a generalization of the Opdam–Cherednik integral operator and its inversion formula on the constructed spaces of Boehmians. Furthermore, it derives various properties of the new pair of generalized Opdam–Cherednik integral operators in a generalized context.

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Acknowledgements

The authors would like to thank the referees for their insightful comments and Springer Nature for its support.

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Correspondence to Shrideh Khalaf Al-Omari.

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Al-Omari, S.K., Araci, S. & Al-Smadi, M. A new structure of an integral operator associated with trigonometric Dunkl settings. Adv Differ Equ 2021, 336 (2021). https://doi.org/10.1186/s13662-021-03485-8

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  • DOI: https://doi.org/10.1186/s13662-021-03485-8

MSC

  • 54C40
  • 14E20
  • 46E25
  • 20C20

Keywords

  • Cherednik–Opdam integral operator
  • Convolution product
  • Polynomial
  • Differential–difference operator
  • Boehmian