An extension of the Bessel–Wright transform in the class of Boehmians

In this paper, we first construct a suitable Boehmian space on which the Bessel–Wright transform can be defined and some desired properties are obtained in the class of Boehmians. Some convergence results are also established.


Introduction
The space of Boehmians is constructed using an algebraic approach that utilizes convolution and approximate identities or delta sequences. If the construction is applied to a function space and the multiplication is interpreted as convolution, the construction yields a space of generalized functions. Those spaces provide a natural setting for extensions of the Bessel-Wright transform newly introduced by Fitouhi et al. [6]. We cite here, as briefly as possible, some facts about harmonic analysis related to the Bessel-Wright operator α,β . For more details, we refer to [6]. We consider, on (0, ∞) the difference differential operator indexed by two parameters α and β (1.1) These operators are very important in pure mathematics and especially in special functions and harmonic analysis. The Bessel-Wright functions j (α, β)(λx) are eigenfunctions of the Bessel-Wright operator α,β with the eigenvalues −λ 2 , (λ ∈ C the space of complex numbers), which is even and symmetric in α and β and coincides when α = 0 or β = 0 with the normalized Bessel function given by The importance of the Bessel-Wright Fourier transform lies in the fact that it generalizes many integral transforms. In fact, given α > 1, the Bessel-Wright Fourier transform reduces essentially to those in [15]: a. The Hankel transform when β = 0. b. The Y transform when β = 1 2 . c. The Hartley transform when β = −1 4 . We denote by: • D(R) the space of test functions with bounded support over R.
• C 0 the space of continuous functions on [0, +∞[ having 0 as limit at infinity.
The Bessel-Wright function is related to the Bessel function via the intertwining operator (Riemann Liouville operator)

Proposition 1.1 The Bessel-Wright transform is related to the Bessel-Fourier transform via
The following two definitions are needed for our results.

Definition 1.2
The Mellin-type convolution product of first kind is defined by:

Definition 1.3
Let α > − 1 2 and f, g ∈ L 1 (0, ∞). Then we define the product ⊗ of f and g by the integral .

Generated spaces of Boehmians
The class of Boehmians was introduced to generalize regular operators [5]. The minimal structure necessary for the abstract construction of Boehmian spaces consists of the following elements: iii. An operation : a × b −→ a such that, for each x ∈ a and s 1 , The elements of are called delta sequences. Denote by Q the set If (x n , s n ), (y n , t n ) ∈ Q, x n t m = y m s n ∀m, n ∈ N, then we say that (x n , s n ) ∼ (y n , t n ). The relation ∼ is an equivalence relation in Q. The space of equivalence classes in Q is denoted by B. The elements of B are called Boehmians. Between a and B, there is a canonical embedding expressed as x → x s n s n .
The relationship between the notion of convergence and the product is specified as follows: The operation is extended to B × b as follows: The convergence in B is defined as follows: Several integral transforms were extended to various spaces of Boehmians by numerous authors, we refer to Karunakaran and Roopkumar [7], Karunakaran and Vembu [8], Mikusinski [11], Al-Omari [1][2][3], Al-Omari and Kilicman [4], Nemzer [13], among others. For a general construction of Boehmians, we refer to [9,10,12].
We establish the following technical result.
Proof Assume that the hypothesis of the theorem is satisfied. Now, on the basis of (1.2) and (1.3), we can write By Fubini's theorem, this can be rewritten as Changing the variables z = xt −1 , we get The spaces generated here are the space We denote by , the set of delta sequences (δ n ) of D(0, ∞) with the following properties: where m is a positive real number The general properties of × are given (see [14,16]) as follows: Let us now establish that B 1 is a Boehmian space. We prefer to omit the proof for B 2 as its details are simlar.

By the change of variables
Hence, The proofs of identities (i) and (iii) follow from simple integral calculus. Identity (ii) directly follows from (2.4).
Thus, using (2.1) in (2.8), we conclude that where [a, b] is an interval containing the support of g y . Therefore, inequality (2.9) implies that From (2.6), (2.7) and (2.10), we get Hence, the equation presented above gives Hence the result.
We define the sum and multiplication by a scalar of two Boehmians in B 1 in the natural way as: and where γ ∈ C.
The operation × and the operation of differentiation are defined as follows: and
The equivalent statement for the δ convergence has the following form: and ϕ n,k → ϕ k as n → ∞ in L 1 α for each k ∈ N.
A sequence of Boehmians (ζ n ) in B 1 is said to be convergent to a Boehmian ζ in B 1 denoted by ζ n − → ζ , if there exists a delta sequence (δ n ) ∈ such that (β n − β) × δ n ∈ L 1 α ∀n ∈ N and (ζ n − ζ ) × δ n → 0 as n → ∞ in L 1 α . Similarly, the following theorems generate the Boehmian space B 2 .
Theorem 2.8 For f ∈ L 1 α (0, ∞) and ψ 1 , ψ 2 ∈ D(0, ∞), the following relation is true: The proofs of Theorems 2.6 and 2.7are similar to the proofs of Theorems 2.2 and 2.3, respectively. The proof of Theorem 2.8 follows from Proposition 1.4. The sum of two Boehmians in B 2 and the operation of multiplication by a scalar can be also defined as follows: and where γ ∈ C.
The operation ⊗ and the operation of differentiation are, respectively, defined as: and The notions of δ-convergence and -convergence in B 1 and B 2 can be defined in a natural way as above.

The Bessel-Wright transform of a Boehmian
Let ζ ∈ B 1 and ζ = [ ( f n ) (δ n ) ]. Then, for every α > − 1 2 we define the generalized Bessel-Wright transform of ζ as follows: The right-hand side of (3.1) belongs to B 2 by virtue of Proposition 1.5. The definition presented above is indeed well defined. Let Then, by the notion of equivalence classes in B 1 , we find f n × ε m = g m × ω n .
Relation (3.1) and the notion of equivalence classes in B 2 yield Hence, we conclude that Thus, we get This proves the claim.
On the other hand, the notion of equivalent classes in B 1 yields We now show that F ge α,β , is a surjective mapping. Let F (α,β) is a surjective mapping. Let (F (α,β) f n ) (ω n ) ∈ B 2 be arbitrary. Then