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Estimates and properties of certain q-Mellin transform on generalized q-calculus theory

Abstract

This paper deals with the generalized q-theory of the q-Mellin transform and its certain properties in a set of q-generalized functions. Some related q-equivalence relations, q-quotients of sequences, q-convergence definitions, and q-delta sequences are represented. Along with that, a new q-convolution theorem of the estimated operator is obtained on the generalized context of q-Boehmians. On top of that, several results and q-Mellin spaces of q-Boehmians are discussed. Furthermore, certain continuous q-embeddings and an inversion formula are also discussed.

Introduction and preliminaries

The quantum calculus or the q-calculus theory has been given a noticeable importance and popularity due to its wide application in various fields of mathematics, statistics, and physics [1]. The q-calculus theory has appeared as a connection between mathematics and physics. Recently, this topic has attracted the attention of several researchers, and a variety of results have been derived in various areas of research including number theory, hypergeometric functions, orthogonal polynomials, quantum theory, combinatorics, and electronics as well. The q-calculus begins with the definition of the q-analogue \(d_{q}g\) of the differential

$$ d_{q}g ( t ) =g ( qt ) -g ( t ) $$

of the function g, where q is a fixed real number such that \(0< q<1\) (see [13]). Having said this, we immediately get the q-analogue of the derivative of g as

$$ D_{q}g ( t ):=\frac{d_{q}g ( t ) }{d_{q}t}:= \frac{g ( t ) -g ( qt ) }{ ( 1-q ) t} \quad \text{for }t\neq 0 $$

and \(D_{q}g ( 0 )=\lim_{t\longrightarrow 0}D_{q}g (t )=g^{{\prime }} ( 0 )\) provided \(g^{{\prime }} ( 0 ) \) exists. Also, when g is differentiable, the q-derivative \(D_{q}g\) tends to \(g^{{\prime }} ( 0 ) \) as q tends to 1. It also satisfies the q-analogue of the Leibniz rule

$$ D_{q} \bigl( g_{1} ( t ) g ( t ) \bigr) =g ( t ) D_{q}g_{1} ( t ) +g_{1} ( qt ) D_{q}g ( t ). $$

The Jackson q-integrals from 0 to x and respectively from 0 to ∞ are defined by [1, 4]

$$\begin{aligned} & \int _{0}^{x}g ( t ) \,d_{q}t= ( 1-q ) t \sum_{0}^{\infty }g \bigl( tq^{k} \bigr) q^{k}, \end{aligned}$$
(1)
$$\begin{aligned} & \int _{0}^{\infty }g ( t ) \,d_{q}t= ( 1-q ) t \sum_{-\infty }^{\infty }g \bigl( q^{k} \bigr) q^{k}, \end{aligned}$$
(2)

when the sums converge absolutely. The Jackson q-integral on the generic interval \([ a,b ] \) is, therefore, given by [1, 5]

$$ \int _{a}^{b}g ( t ) \,d_{q}t= \int _{0}^{b}g ( t ) \,d_{q}t- \int _{0}^{a}g ( t ) \,d_{q}t. $$

The q-integration by parts for two functions f and g is defined by

$$ \int _{0}^{b}g_{2} ( t ) D_{q}g_{1} ( t ) \,d_{q}t=g_{1} ( b ) g_{2} ( b ) -g_{1} ( a ) g_{2} ( a ) - \int _{a}^{b}g_{1} ( qt ) D_{q}g_{2} ( t ) \,d_{q}t. $$

Arising from the notion of regular operators [6], the notion of a Boehmian was firstly introduced by Mikusinski and Mikusinski [7] to generalize distributions and regular operators [8]. Boehmians are equivalence classes of quotients of sequences constructed from an integral domain when the operations are interpreted as addition and convolution, see, e.g., [920]. In terms of the q-calculus concept, we introduce the concept of q-Boehmians to popularize the concept of q-calculus theory as follows:

For a complex linear space V and a subspace \(( W,\ast ^{q} ) \) of V, let \(\overset{q}{\bullet }:V\times W\rightarrow V\) be a binary operation such that the undermentioned axioms (1)–(5) hold:

  1. (1)

    \(( g_{1}+g_{2} ) \overset{q}{\bullet } \psi =g_{1}\overset{q}{\bullet }\psi +g_{2}\overset{q}{\bullet }\psi,\forall g_{1},g_{2} \in V\text{ and }\psi \in W\).

  2. (2)

    \(( \alpha g ) \overset{q}{\bullet }\psi = \alpha ( g\overset{q}{\bullet }\psi ),\forall \alpha \in \mathbb{C},\forall g\in V\text{ and }\psi \in W\).

  3. (3)

    \(g\overset{q}{\bullet } ( \psi _{1} \overset{q}{\bullet }\psi _{2} ) = ( g\overset{q}{\bullet }\psi _{1} ) \overset{q}{\bullet }\psi _{2},\forall g\in V\text{ and }\psi _{1},\psi _{2}\in W\).

  4. (4)
    $$\begin{aligned} & \text{If }g_{n}\rightarrow g\text{ in }V\text{ as }n\rightarrow \infty \text{ and }\psi \in W, \\ &\quad\text{then }g_{n} \overset{q}{\bullet }\psi \rightarrow g\overset{q}{ \bullet }\psi \text{ as }n \rightarrow \infty \text{ in }V. \end{aligned}$$
    (3)
  5. (5)

    A collection \(\Delta _{q}\) of sequences from W such that, for all \(( \varepsilon _{n} ), ( \phi _{n} ) \in \Delta _{q}\) and \(( g_{n} ) \in W\), we have \(\varepsilon _{n}\overset{q}{\bullet }\phi _{n}\in \Delta _{q}\) and

    $$ \text{if }g_{n}\rightarrow g\text{ in }V\text{ as }n\rightarrow \infty,\text{ then }g_{n}\overset{q}{\bullet }\varepsilon _{n}\rightarrow g \text{ as }n\rightarrow \infty. $$

Once the preceding axioms are applied, the name of a q-Boehmian is set to mean the equivalence class \(\frac{g_{n}}{\delta _{n}}\) that arises from the equivalence relation

$$ g_{n}\overset{q}{\bullet }\varepsilon _{m}=g_{m} \overset{q}{\bullet }\varepsilon _{n},\quad\forall m,n\in \mathbb{N}, $$
(4)

where \(( g_{n} ) \in V\) and \(( \varepsilon _{n} ) \in \Delta _{q}\). The collection of all q-Boehmians is denoted by \(\mathbb{B}_{q}\) which is the so-called Boehmian space. The classical linear space V is identified as a subset of the space \(\mathbb{B}_{q}\) which can be recognized from the relation

$$ g\longrightarrow \frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}}, $$
(5)

where \(( \varepsilon _{n} ) \in \Delta _{q}\) is arbitrary. Two q-Boehmians \(\frac{g_{n}}{\varepsilon _{n}}\) and \(\frac{\varphi _{n}}{\epsilon _{n}}\) are said to be equal in \(\mathbb{B}_{q}\) if \(g_{n}\overset{q}{\bullet }\epsilon _{m}=\varphi _{m}\overset{q}{\bullet }\varepsilon _{n}, \forall m,n\in \mathbb{N} \). Addition in the space \(\mathbb{B}_{q}\) is defined as

$$ \frac{g_{n}}{\varepsilon _{n}}+\frac{\varphi _{n}}{\epsilon _{n}}= \frac{g_{n}\overset{q}{\bullet }\epsilon _{n}+\varphi _{n}\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}\overset{q}{\bullet }\epsilon _{n}}. $$
(6)

The scalar multiplication in the space \(\mathbb{B}_{q}\) is defined as

$$ \alpha \frac{g_{n}}{\varepsilon _{n}}= \frac{\alpha g_{n}}{\varepsilon _{n}},\quad \alpha \in \mathbb{C}. $$

The q-convergence of type δ, \(\beta _{n}\overset{\delta }{\rightarrow }\beta \), is defined in the space \(\mathbb{B}_{q}\) when for \(( \psi _{n} ) \in \Delta _{q}\) and each \(k\in \mathbb{N} \) such that

$$ \beta _{n}\overset{q}{\bullet }\varepsilon _{k}\in V,\quad \forall k,n \in \mathbb{N}, \beta \overset{q}{\bullet }\varepsilon _{k} \in V, $$
(7)

we have \(\beta _{n}\overset{q}{\bullet }\varepsilon _{k}\rightarrow \beta \overset{q}{\bullet }\varepsilon _{k}\) as \(n\rightarrow \infty \) in V. The q-convergence \(\beta _{n}\overset{\Delta _{q}}{\rightarrow }\beta \) of type \(\Delta _{q}\) is defined when for some \(( \varepsilon _{n} ) \in \Delta _{q}\) we have

$$ ( \beta _{n}-\beta ) \overset{q}{\bullet }\varepsilon _{n} \in V,\quad \forall n\in \mathbb{N} \quad\text{and}\quad ( \beta _{n}- \beta ) \overset{q}{\bullet }\varepsilon _{n}\rightarrow 0\quad\text{as }n \rightarrow \infty \text{ in }V. $$
(8)

The space of q-Boehmians emerging from the q-convergence assigns a complete quasi-normed space.

In recent work, several remarkable integral transforms were given different q-analogues in a q-calculus context [4, 2124]. In the sequence of such q-integral transforms, we recall the q-Laplace integral transform [2529], the q-Sumudu integral transform [2, 3032], the q-Weyl fractional integral transform [33], the q-wavelet integral transform [34], the q-Mellin type integral transform [35], the Mangontarum integral transform [36, 37], the \(E_{2;1}\) integral transform [38, 39], the natural integral transform [3], and many others, to mention but a few. In this paper, we discuss the generalized q-theory of the q-Mellin transform and obtain several results.

Let g be a function defined on \(\mathbb{R}_{q,+},\mathbb{R}_{q,+}=\{q^{n}:n\in \mathbb{Z}\}\), then the q-Mellin transform was defined by [40], p. 521 as

$$ M_{q} \bigl( g ( t ) \bigr) ( \zeta ) = \int _{0}^{\infty }t^{\zeta -1}g ( t ) \,d_{q}t, $$
(9)

provided the q-integral converges. The integral (9) is analytic on the fundamental strip \(\langle \alpha _{q,g};\beta _{q,g} \rangle \) and periodic with period \(2i\pi \log ( q ) \). The inversion formula for the q-analogue (9) is given by

$$ g(t)=\frac{\log (q)}{2i\pi (1-q)} \int _{c- \frac{i\pi }{\log (q)}}^{c+\frac{i\pi }{\log (q)}}M_{q}(g) (\zeta )t^{-\zeta }\,d\zeta, \quad t\in \mathbb{R}_{q,+}, $$

where \(\alpha _{q,g}< c<\beta _{q,g}\). The asymptotic properties as well as the asymptotic singularities of the q-Mellin transform into asymptotic expansions of the original function for \(x\rightarrow 0\) and \(x\rightarrow \infty \) are given in [40]. Additionally, the asymptotic behavior at 0 or ∞ is studied using the q-Mellin transform.

Definition 1

The function g is said to be q-integrable on an interval \([0,\infty[ \) provided the infinite series

$$ \sum_{n\in \mathbb{Z}}q^{n}g \bigl(q^{n} \bigr) $$

converges absolutely. The space of all q-integrable functions on \([0,\infty[ \) is denoted by \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \). In a better recognition, the space \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) is defined to be the space of all q-integrable functions g on \(\mathbb{R}_{q,+}\) such that

$$ L_{q}^{1}g ( t ) =\frac{1}{1-q} \int _{0}^{ \infty } \bigl\vert g ( t ) \bigr\vert \,d_{q}t< \infty. $$
(10)

We denote by \(\mathbb{D}_{q}\) the q-space of test functions of compact supports on \(\mathbb{R}_{q,+}\), i.e., \(\mathbb{D}_{q}\) is the q-space of all smooth functions \(\kappa \in C^{\infty } ( \mathbb{R}_{q,+} ) \) such that

$$ \mathbb{D}_{q}= \Bigl\{ \kappa \in C^{\infty } ( \mathbb{R}_{q,+} ):\sup_{0< t< \infty } \bigl\vert D_{q}\kappa ( t ) \bigr\vert < \infty \Bigr\} . $$
(11)

However, this theory is new and might be developing a new area of research. It investigates a generalization to the q-theory of calculus [40] and hence all results can be popularized. Every element in the space \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) is identified as a member in the generalized theory. To this aim, we spread our results into five sections. In Sect. 1, we recall some definitions and preliminaries from the q-calculus theory. In Sect. 2, we derive q-delta sequences, q-convolution theorems and establish a space of q-Boehmians. In Sect. 3, we establish a space of q-ultraBoehmians. In Sect. 4, we generalize definitions and obtain several properties of the q-Mellin transform. In Sect. 5 we include several results.

The space \(\mathbb{B}\)

In this section, we strive to establish the space \(\mathbb{B}\) of q-Boehmians. Henceforth, we denote by \(\Delta _{q}\) the set of all sequences from \(\mathbb{D}_{q}\) such that the undermentioned identities \(\Delta _{q}^{1}-\Delta _{q}^{3}\) hold, where

$$ \begin{aligned}& \Delta _{q}^{1}: \int _{0}^{\infty } \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t=1,\quad \forall n\in \mathbb{N}, \\ &\Delta _{q}^{2}: \bigl\vert \varepsilon _{n} ( t ) \bigr\vert < M,\quad M>0,M\in \mathbb{R} _{+}, \\ &\Delta _{q}^{3}:\operatorname{supp}(\varepsilon _{n})\subseteq ( 0,b_{n} ),\quad b_{n}\rightarrow 0 \text{ as }n\rightarrow \infty,0< b_{n}, \forall n\in \mathbb{N}.\end{aligned} $$
(12)

On the other hand, we denote by \(\overset{q}{\bullet }\) the Mellin type q-convolution product defined on \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) by

$$ ( g_{1}\overset{q}{\bullet }g_{2} ) ( x ) = \int _{0}^{\infty }t^{-1}g_{1} \bigl( t^{-1}x \bigr) g_{2} ( t ) \,d_{q}t, $$
(13)

provided the integral part exists for every \(x>0\). It is clear from the context that \(g_{1}\overset{q}{\bullet }g_{2}\in L_{q}^{1} ( \mathbb{R}_{q,+} ) \) for all \(g_{1}\) and \(g_{2}\) in \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \). On that account, the q-convolution theorem of the q-Mellin transform of the product \(g_{1}\overset{q}{\bullet }g_{2}\) can be easily established as follows.

Theorem 2

Let \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) be the space of all q-integrable functions on \(\mathbb{R}_{q,+}\). Then the q-convolution theorem of the transform \(M_{q}\) is given by

$$ M_{q} ( g_{1}\overset{q}{\bullet }g_{2} ) ( \zeta ) =M_{q}g_{1} ( \zeta ) M_{q}g_{2} ( \zeta ) \quad\textit{for }g_{1}\textit{ and }g_{2} \textit{ in }L_{q}^{1} ( \mathbb{R}_{q,+} ). $$

Proof

By applying the definition of the \(M_{q}\) transform to the product \(g_{1}\overset{q}{\bullet }g_{2}\), we get

$$\begin{aligned} M_{q} ( g_{1}\overset{q}{\bullet }g_{2} ) ( \zeta ) &= \int _{0}^{\infty } ( g_{1} \overset{q}{ \bullet }g_{2} ) ( x ) x^{\zeta -1}\,d_{q}x \\ &= \int _{0}^{\infty } \biggl( \int _{0}^{\infty }g_{1} ( t ) g_{2} \bigl( t^{-1}x \bigr) x^{-1}\,d_{q}t \biggr) x^{ \zeta -1}\,d_{q}x. \end{aligned}$$

Therefore, employing the substitution \(z=t^{-1}x\) and, hence, \(d_{q}z=t^{-1}\,d_{q}x\), in collaboration with simple computations, reveals

$$ M_{q} ( g_{1}\overset{q}{\bullet }g_{2} ) ( \zeta ) =M_{q} ( g_{1} ) ( \zeta ) M_{q} ( g_{2} ) ( \zeta ). $$

Hence, the proof of this theorem is completed. □

The following is an imperative result for initiating the q-delta sequence concept.

Lemma 3

Let \(( \varepsilon _{n} ) \) and \(( \epsilon _{n} ) \) be sequences in \(\Delta _{q}\). Then \(( \varepsilon _{n}\overset{q}{\bullet }\epsilon _{n} ) \) is a sequence in \(\Delta _{q}\).

Proof

To establish this lemma, we examine the performance of the sequence \(( \varepsilon _{n}\overset{q}{\bullet }\epsilon _{n} ) \). To inspect the correctness of the property \(\Delta _{q}^{1}\), we use the integral equation (3) to get

$$ \int _{0}^{\infty } ( \varepsilon _{n} \overset{q}{\bullet }\epsilon _{n} ) ( x ) \,d_{q}x= \int _{0}^{ \infty }t^{-1}\epsilon _{n} ( t ) \biggl( \int _{0}^{ \infty }\varepsilon _{n} \bigl( t^{-1}x \bigr) \,d_{q}x \biggr) \,d_{q}t. $$
(14)

Therefore, by using the change of variables \(t^{-1}x=y\) and, hence, \(d_{q}x=t\,d_{q}y\), (14) we indicate

$$ \int _{0}^{\infty } ( \varepsilon _{n} \overset{q}{\bullet }\epsilon _{n} ) ( x ) \,d_{q}x= \biggl( \int _{0}^{\infty }\epsilon _{n} ( t ) \,d_{q}t \biggr) \biggl( \int _{0}^{\infty }\varepsilon _{n} ( y ) \,d_{q}y \biggr) =1. $$

This proves the \(\Delta _{q}^{1}\) part. The proof of the \(\Delta _{q}^{2}\) part follows from similar techniques, whereas the \(\Delta _{q}^{3}\) part is clearly valid, by conducting the fact

$$ \operatorname{supp} ( \varepsilon _{n}\overset{q}{\bullet }\epsilon _{n} ) \subset \operatorname{supp} ( \varepsilon _{n} ) + \operatorname{supp} ( \epsilon _{n} ) \quad\text{for } ( \varepsilon _{n} ), ( \epsilon _{n} ) \in \Delta _{q}. $$

This ends the proof of the lemma. □

Lemma 3, hence, displays that every sequence in \(\Delta _{q}\) forms, to a great extent, the q-delta sequence concept.

Lemma 4

Let \(g_{1},g_{2}\in L_{q}^{1} ( \mathbb{R}_{q,+} ) \), \(\kappa _{1},\kappa _{2}\in \mathbb{D}_{q}\), and \(\alpha \in \mathbb{C}\). Then the following assertions are valid:

$$\begin{aligned} &\mathrm{(i)}\quad \kappa _{1}\overset{q}{\bullet }\kappa _{2}=\kappa _{2}\overset{q}{\bullet }\kappa _{1},\qquad \mathrm{(ii)}\quad ( g_{1}+g_{2} ) \overset{q}{ \bullet }\kappa _{1}=g_{1}\overset{q}{\bullet } \kappa _{1}+g_{2}\overset{q}{\bullet }\kappa _{1}, \\ &\mathrm{(iii)}\quad ( \alpha g_{1} ) \overset{q}{\bullet }\kappa _{1}= \alpha ( g_{1} \overset{q}{\bullet }\kappa _{1} ),\qquad \mathrm{(iv)}\quad g_{1} \overset{q}{\bullet } ( \kappa _{1}\overset{q}{ \bullet }\kappa _{2} ) = ( g_{1}\overset{q}{ \bullet }\kappa _{1} ) \overset{q}{\bullet }\kappa _{2}. \end{aligned}$$

Proof

(i) As the convolution product of the functions \(\kappa _{1}\) and \(\kappa _{2}\) in \(\mathbb{D}_{q}\) is exceptionally given by

$$ ( \kappa \overset{q}{\bullet }\kappa _{2} ) ( x ) = \int _{0}^{\infty }t^{-1}\kappa _{1} \bigl( t^{-1}x \bigr) \kappa _{2} ( t ) \,d_{q}t, $$
(15)

the change of variables \(t^{-1}x=y\) reveals us to write (15) into the form

$$ ( \kappa _{1}\overset{q}{\bullet }\kappa _{2} ) ( x ) = \int _{0}^{\infty }y^{-1}\kappa _{2} \bigl( x^{-1}y \bigr) \kappa _{1} ( y ) \,d_{q}y. $$

Hence (i) follows. To prove (ii) and (iii), we merely follow simple integral calculus. To prove (iv), we employ the definition of the product \(\overset{q}{\bullet }\) to get

$$\begin{aligned} \bigl( g_{1}\overset{q}{\bullet } ( \kappa _{1} \overset{q}{\bullet }\kappa _{2} ) \bigr) ( x ) &= \int _{0}^{ \infty }t^{-1}g_{1} \bigl( t^{-1}x \bigr) ( \kappa _{1} \overset{q}{\bullet }\kappa _{2} ) ( t ) \,d_{q}t \\ &= \int _{0}^{\infty }t^{-1}g_{1} \bigl( t^{-1}x \bigr) \biggl( \int _{0}^{\infty }y^{-1}\kappa _{1} \bigl( y^{-1}t \bigr) \kappa _{2} ( y ) \,d_{q}y \biggr) \,d_{q}t. \end{aligned}$$

That is,

$$ \bigl( g_{1}\overset{q}{\bullet } ( \kappa _{1} \overset{q}{\bullet }\kappa _{2} ) \bigr) ( x ) = \int _{0}^{ \infty }y^{-1} \biggl( \int _{0}^{\infty }t^{-1}g_{1} \bigl( t^{-1}x \bigr) \kappa _{1} \bigl( y^{-1}t \bigr) \,d_{q}t \biggr) \kappa _{2} ( y ) \,d_{q}y. $$
(16)

Now, by employing the change of variables \(y^{-1}t=z\), we write down equation \(( 16 ) \) into the form

$$\begin{aligned} \bigl( g_{1}\overset{q}{\bullet } ( \kappa _{1} \overset{q}{\bullet }\kappa _{2} ) \bigr) ( x ) &= \int _{0}^{ \infty }y^{-1} \biggl( \int _{0}^{\infty }z^{-1}g_{1} \bigl( z^{-1} \bigl( y^{-1}x \bigr) \bigr) \kappa _{1} ( z ) \,d_{q}z \biggr) \kappa _{2} ( y ) \,d_{q}y \\ &= \int _{0}^{\infty }y^{-1} ( g_{1} \overset{q}{\bullet }\kappa _{1} ) \bigl( y^{-1}x \bigr) \kappa _{2} ( y ) \,d_{q}y. \end{aligned}$$

This ends the proof of the lemma. □

To proceed in our construction, we establish the following lemma.

Lemma 5

(i) Let \(g_{1}\) and \(g_{2}\) be integrable functions in \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) and \(( \varepsilon _{n} ) \) be a delta sequence in the set \(\Delta _{q}\) such that \(g_{1}\overset{q}{\bullet }\varepsilon _{n}=g_{2} \overset{q}{\bullet }\varepsilon _{n}\). Then \(g_{1}=g_{2}\) in \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) for every \(n\in \mathbb{N}\).

(ii) Let g and \(( g_{n} ) \) be integrable functions in \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) such that \(g_{n}\rightarrow g\) as \(n\rightarrow \infty \) in \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \). Then

$$ g_{n}\overset{q}{\bullet }\psi \rightarrow g\overset{q}{\bullet } \psi \quad\textit{for every }\psi \in \mathbb{D}_{q}\textit{ as }n \rightarrow \infty. $$

Proof

To prove (i), we merely need to show that \(g_{1}\overset{q}{\bullet }\varepsilon _{n}=g_{1}\in L_{q}^{1} ( \mathbb{R}_{q,+} ) \). By using \(\Delta _{q}^{1}\) and \(\Delta _{q}^{3}\), we obtain

$$\begin{aligned} \int _{0}^{\infty } \bigl\vert ( g_{1} \overset{q}{\bullet }\varepsilon _{n} ) ( x ) -g_{1} ( x ) \bigr\vert \,d_{q}x &\leq \int _{0}^{\infty } \int _{0}^{\infty } \bigl\vert t^{-1}g_{1} \bigl( t^{-1}x \bigr) -g_{1} ( x ) \bigr\vert \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t \,d_{q}x \\ &= \int _{0}^{\infty } \int _{a_{n}}^{b_{n}} \bigl\vert t^{-1}g_{1} \bigl( t^{-1}x \bigr) -g_{1} ( x ) \bigr\vert \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t \,d_{q}x. \end{aligned}$$

Therefore,

$$\begin{aligned} & \int _{0}^{\infty } \bigl\vert ( g_{1} \overset{q}{\bullet }\varepsilon _{n} ) ( x ) -g_{1} ( x ) \bigr\vert \,d_{q}x \\ &\quad \leq \int _{0}^{\infty } \int _{a_{n}}^{b_{n}} \bigl\vert t^{-1}g_{1} \bigl( t^{-1}x \bigr) \bigr\vert \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t\,d_{q}x \\ &\qquad{}+ \int _{0}^{\infty } \int _{a_{n}}^{b_{n}} \bigl\vert g_{1} ( x ) \bigr\vert \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t\,d_{q}x. \end{aligned}$$
(17)

Hence, for \(g_{1}\in L_{q}^{1} ( \mathbb{R}_{q,+} ) \), by using (17) we turn to write

$$ \int _{0}^{\infty } \bigl\vert ( g_{1} \overset{q}{\bullet }\varepsilon _{n} ) ( x ) -g_{1} ( x ) \bigr\vert \,d_{q}x\leq A \int _{0}^{b_{n}} \bigl\vert t^{-1} \bigr\vert \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t+A \int _{0}^{b_{n}} \bigl\vert \varepsilon _{n} ( t ) \bigr\vert \,d_{q}t. $$

Therefore, by the properties of the delta sequences \(\Delta _{q}^{2}\) and \(\Delta _{q}^{3}\), we conclude that

$$ \int _{0}^{\infty } \bigl\vert ( g_{1} \overset{q}{\bullet }\varepsilon _{n} ) ( x ) -g_{1} ( x ) \bigr\vert \,d_{q}x\leq AM\ln ( b_{n} ) +AM ( b_{n} ) \rightarrow 0 $$

as \(n\rightarrow \infty \).

Proof of (ii) follows from simple integration. We therefore omit the details. Hence the proof of this lemma is ended. □

Lemma 6

Let \(g_{1}\) be an integrable function in the space \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \). Then \(g_{1}\overset{q}{\bullet }\varepsilon _{n}\rightarrow g_{1}\) as \(n\rightarrow \infty \) for every \(( \varepsilon _{n} ) \in \Delta _{q}\).

The proof of this lemma is a straightforward conclusion from the proof of Lemma 4. Hence, we delete the details.

Thus, the space \(\mathbb{B}\) with \(( L_{q}^{1} ( \mathbb{R}_{q,+} ),\overset{q}{\bullet } ), ( \mathbb{D}_{q}, \overset{q}{\bullet } ) \), and \(\Delta _{q}\) is defined. The canonical embedding of \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) in \(\mathbb{B}\) is given by

$$ g\rightarrow \frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}}. $$
(18)

That is, every element in the space \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) can be identified as a member of the space \(\mathbb{B}\). Addition, scalar multiplication, differentiation, \(\Delta _{q}\) and \(\delta _{q}\) convergence are defined in a natural way as follows:

If \(( \varphi _{n} ) \in L_{q}^{1} ( \mathbb{R}_{q,+} ) \) and \(( \varepsilon _{n} ) \in \Delta _{q}\), then the pair \(( \varphi _{n},\varepsilon _{n} ) \) \(( \text{or }\frac{\varphi _{n}}{\varepsilon _{n}} ) \) is said to be a q-quotient of sequences if \(\varphi _{n}\overset{q}{\bullet }\varepsilon _{m}=\varphi _{m} \overset{q}{\bullet }\varepsilon _{n},\forall n,m\in \mathbb{N}\). Therefore, if \(\frac{\varphi _{n}}{\epsilon _{n}}\) and \(\frac{g_{n}}{\varepsilon _{n}}\) are q-quotients of sequences and \(g\in L_{q}^{1} ( \mathbb{R}_{q,+} ) \), then it is easy to see that

$$ \frac{g\overset{q}{\bullet }\epsilon _{n}}{\epsilon _{n}}\quad\text{and}\quad \frac{\varphi _{n}\overset{q}{\bullet }\epsilon _{n}+g_{n}\overset{q}{\bullet }\epsilon _{n}}{\epsilon _{n}\overset{q}{\bullet }\varepsilon _{n}} $$

are q-quotients of sequences. Two q-quotients of sequences \(\frac{\varphi _{n}}{\epsilon _{n}}\) and \(\frac{g_{n}}{\varepsilon _{n}}\) are said to be equivalent if

$$ \varphi _{n}\overset{q}{\bullet }\varepsilon _{m}=g_{m} \overset{q}{\bullet }\epsilon _{n},\quad \forall n,m\in \mathbb{N}. $$

We can easily check the following equivalence relations:

$$ \frac{\varphi _{n}}{\epsilon _{n}\overset{q}{\bullet }g}\sim \frac{\varphi _{n}\overset{q}{\bullet }g}{\epsilon _{n}}\quad\text{and}\quad \frac{\varphi _{n}}{\epsilon _{n}\overset{q}{\bullet }g_{n}} \sim \frac{\varphi _{n}\overset{q}{\bullet }g_{n}}{\epsilon _{n}}. $$

The equivalent class \(\breve{w}= ( \frac{\varphi _{n}}{\epsilon _{n}} ) \) of q-quotients of sequences containing \(\frac{\varphi _{n}}{\epsilon _{n}}\) is said to be a q-Boehmian. The space of such q-Boehmians is denoted by \(\mathbb{B}\).

Remark 7

For two q-Boehmians \(\breve{w}= ( \frac{\varphi _{n}}{\epsilon _{n}} ) \) and \(\breve{z}= ( \frac{g_{n}}{\varepsilon _{n}} ) \) in \(\mathbb{B}\), we have the following identities:

$$\begin{aligned} &\mathrm{(i)}\quad \breve{w}+\breve{z}= \biggl( \frac{\varphi _{n}\overset{q}{\bullet }\epsilon _{n}+g_{n}\overset{q}{\bullet }\epsilon _{n}}{\epsilon _{n}\overset{q}{\bullet }\varepsilon _{n}} \biggr), \\ &\mathrm{(ii)}\quad \beta \breve{w}= \biggl( \frac{\beta \varphi _{n}}{\epsilon _{n}} \biggr), \\ &\mathrm{(iii)}\quad \breve{w}\overset{q}{\bullet }\breve{z}= \biggl( \frac{\varphi _{n}\overset{q}{\bullet }g_{n}}{\epsilon _{n}\overset{q}{\bullet }\varepsilon _{n}} \biggr), \\ &\mathrm{(iv)}\quad D^{k}\breve{w}= \biggl( \frac{D^{k}\varphi _{n}}{\epsilon _{n}} \biggr), \\ &\mathrm{(v)}\quad \breve{w}\overset{q}{\bullet }g= \biggl( \frac{\varphi _{n}\overset{q}{\bullet }g}{\epsilon _{n}} \biggr), \end{aligned}$$

where \(k\in \mathbb{R}\), \(\beta \in \mathbb{C}\) and \(D^{k}\breve{w}\) is the kth derivative of , and \(\psi \in L_{q}^{1} ( \mathbb{R}_{q,+} ) \).

Definition 8

(i) For \(n=1,2,3,\ldots \) and \(\breve{w}_{n},\breve{w}\in \mathbb{B}\), the sequence \(( \breve{w}_{n} ) \) is \(\delta _{q}\)-convergent to , denoted by \(\delta _{q}-\lim_{n\rightarrow \infty }\breve{w}_{n}=\breve{w}\), provided there can be found a q-delta sequence \(( \epsilon _{n} ) \) such that

$$ ( \breve{w}_{n}\overset{q}{\bullet }\epsilon _{k} ), ( \breve{w}\overset{q}{\bullet }\epsilon _{k} ) \quad\text{in }L_{q}^{1} ( \mathbb{R}_{q,+} ) \quad \text{and}\quad\lim_{n \rightarrow \infty }\breve{w}_{n}\overset{q}{ \bullet } \epsilon _{k}= \breve{w}\overset{q}{\bullet } \epsilon _{k}\quad\text{in }L_{q}^{1} ( \mathbb{R}_{q,+} )\ ( \forall \text{ }k\in \mathbb{N} ). $$

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

$$ ( \breve{w}_{n}-\breve{w} ) \overset{q}{\bullet } \epsilon _{n}\in L_{q}^{1} ( \mathbb{R}_{q,+} )\quad ( \forall n\in \mathbb{N} ) \quad\text{and}\quad\lim _{n\rightarrow \infty } ( \breve{w}_{n}- \breve{w} ) \overset{q}{\bullet }\epsilon _{n}=0 \quad\text{in }L_{q}^{1} ( \mathbb{R}_{q,+} ). $$

Now we have the following few corollaries.

Corollary 9

(i) Let \(g\in L_{q}^{1} ( \mathbb{R}_{q,+} ) \) and \(( \epsilon _{n} ) \in \Delta _{q}\) be fixed. Then the mapping

$$ g\rightarrow \breve{w}, $$

where \(\breve{w}=\frac{g\overset{q}{\bullet }\epsilon _{n}}{\epsilon _{n}}\) is an injective mapping from \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) into \(\mathbb{B}\).

(ii) Let \(( \epsilon _{n} ) \in \Delta _{q}\). Then, if \(g_{n}\rightarrow g\) in \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) as \(n\rightarrow \infty \), then for all \(k\in \mathbb{N}\),

$$ g_{n}\overset{q}{\bullet }\epsilon _{k}\rightarrow g \overset{q}{\bullet }\epsilon _{k}\quad\textit{and}\quad \breve{w}_{n}\rightarrow \breve{w} \quad\textit{in }\mathbb{B} \textit{ as }n\rightarrow \infty. $$

Therefore, it can be easily checked that \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) can be mathematically identified as a subspace of \(\mathbb{B}\).

The above corollary leads to the following corollary.

Corollary 10

The q-embedding, \(g\rightarrow \breve{w}\), \(\breve{w}=\frac{g\overset{q}{\bullet }\epsilon _{n}}{\epsilon _{n}}\), of the space \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) into the space \(\mathbb{B}\) is continuous.

The q-ultraBoehmian space \(\mathbb{B}_{\mathbb{M}}\)

In this section, we provide sufficient axioms to define the q-ultraBoehmian space \(\mathbb{B}_{\mathbb{M}}\) with the set \(( L_{\mathbb{M}},\circ ) \), the subset \(( \mathbb{D}_{\mathbb{M}},\circ ) \), the set \(( \Delta _{q,\mathbb{M}},\circ ) \) of q-delta sequences, and the product , where \(L_{\mathbb{M}},\mathbb{D}_{\mathbb{M}}\), and \(\Delta _{q,\mathbb{M}}\) are the q-Mellin transforms of the sets \(L_{q}^{1} ( \mathbb{R} ),\mathbb{D}_{q}\), and \(\Delta _{q}\) respectively. To this end, we introduce the following convolution operation.

Definition 11

Let \(\omega _{1}\) and \(\omega _{2}\) be in \(\mathbb{B}_{\mathbb{M}}\). For \(\omega _{1}\) and \(\omega _{2}\), we define a product as

$$ ( \omega _{1}\circ \omega _{2} ) ( t ) = \omega _{1} ( t ) \omega _{2} ( t ). $$
(19)

The following assertion holds in the space \(L_{\mathbb{M}}\).

Theorem 12

Let \(\omega _{1}\) be in \(L_{\mathbb{M}}\). Then \(\omega _{1}\circ \eta \in L_{\mathbb{M}}\) for all \(\eta \in \mathbb{D}_{\mathbb{M}}\).

Proof

Let \(\omega _{1}\in L_{\mathbb{M}}\). Then, by the definition of the space \(L_{\mathbb{M}}\) and the definition of the product , we write

$$ ( \omega _{1}\circ \omega _{2} ) ( t ) = \omega _{1} ( t ) \omega _{2} ( t ) =M_{q} ( g_{1} ) M_{q} ( g_{2} ) $$
(20)

for some \(g_{1},g_{2}\in L_{q}^{1} ( \mathbb{R}_{q,+} ) \). Hence, by virtue of Def. 11, (20) can be written in the form

$$ ( \omega _{1}\circ \omega _{2} ) ( t ) =M_{q} ( g_{1}\overset{q}{\bullet }g_{2} ). $$
(21)

Therefore, as \(g_{1}\circ g_{2}\in L_{q}^{1} ( \mathbb{R}_{q,+} ) \), it follows from (21) that \(\omega _{1}\circ \eta \in L_{\mathbb{M}}\). This ends the proof of the theorem. □

Theorem 13

Let ω be an integrable function in \(\mathbb{L}_{\mathbb{M}}\). Then \(\omega \circ ( \eta _{1}\circ \eta _{2} ) = ( \omega \circ \eta _{1} ) \circ \eta _{2}\) for all \(\eta _{1},\eta _{2}\in \mathbb{D}_{q}\).

Proof

By the concept of the convolution , we get

$$ \bigl( \omega \circ ( \eta _{1}\circ \eta _{2} ) \bigr) ( t ) =\omega ( t ) ( \eta _{1}\circ \eta _{2} ) ( t ) = \omega ( t ) \eta _{1} ( t ) \eta _{2} ( t ). $$

By using Def. 11 twice, we write the preceding equation as

$$ \bigl( \omega \circ ( \eta _{1}\circ \eta _{2} ) \bigr) ( t ) = ( \omega \circ \eta _{1} ) ( t ) \eta _{2} ( t ) = \bigl( ( \omega \circ \eta _{1} ) \circ \eta _{2} \bigr) ( t ). $$

This ends the proof of the theorem. □

The following axioms are straightforward.

Theorem 14

(i) Let \(\omega _{1}\) and \(\omega _{2}\) be in \(L_{\mathbb{M}}\). Then \(( \omega _{1}+\omega _{2} ) \circ \eta =\omega _{1} \circ \eta +\omega _{2}\circ \eta \) for all \(\eta \in \mathbb{D}_{q}\).

(ii) Let \(\omega _{1}\) be in \(L_{\mathbb{M}}\). Then \(( \alpha \omega _{1}\circ \eta ) =\alpha ( \omega _{1}\circ \eta ) \) for all \(\eta \in \mathbb{D}_{q}\) and \(\alpha \in \mathbb{C}\).

Proof

(i) Let \(\omega _{1}\) and \(\omega _{2}\) be in \(L_{\mathbb{M}}\). Then, by Def. 11, we write

$$ \bigl( ( \omega _{1}+\omega _{2} ) \circ \eta \bigr) ( t ) = ( \omega _{1}+\omega _{2} ) ( t ) \eta ( t ) =\omega _{1} ( t ) \eta ( t ) +\omega _{2} ( t ) \eta ( t ) = ( \omega _{1}\circ \eta ) ( t ) + ( \omega _{2}\circ \eta ) ( t ). $$

The proof of the first part is finished. The proof of the second part is trivial. This completes the proof of the theorem. □

Theorem 15

(i) Let \(\omega _{1}\) and \(( \omega _{n} ) \) be members of the space \(L_{\mathbb{M}}\) and \(\eta \in \mathbb{D}_{\mathbb{M}}\). If \(\omega _{n}\rightarrow \omega _{1}\) in \(L_{\mathbb{M}}\) as \(n\rightarrow \infty \), then \(\omega _{n}\circ \eta \rightarrow \omega _{1}\circ \eta \) as \(n\rightarrow \infty \).

(ii) Let \(\omega _{1}\) and \(\omega _{2}\) be in \(L_{\mathbb{M}} \) and \(( \upsilon _{n} ) \in \Delta _{q,\mathbb{M}}\). If \(\omega _{1}\circ \upsilon _{n}=\omega _{2}\circ \upsilon _{n}\), then \(\omega _{1}=\omega _{2}\) in \(L_{\mathbb{M}}\).

(iii) Let \(\omega _{1}\) be an integrable function in \(L_{\mathbb{M}}\) and \(( \upsilon _{n} ) \in \Delta _{q,\mathbb{M}},\upsilon _{n} ( t ) \neq 0\) for all \(t\in \mathbb{R}_{q,+}\). Then \(\omega _{1}\circ \upsilon _{n}\rightarrow 0\) in \(L_{\mathbb{M}}\) as \(n\rightarrow \infty \).

Proof

To prove (i), let \(\omega _{1}\) and \(( \omega _{n} ) \) be members of \(L_{\mathbb{M}}\) and \(\eta \in \mathbb{D}_{\mathbb{M}}\). If \(\omega _{n}\rightarrow \omega _{1}\) in \(L_{\mathbb{M}}\) as \(n\rightarrow \infty \), then by Def. 11 and Theo. 14, we have

$$ ( \omega _{n}\circ \eta -\omega _{1}\circ \eta ) ( t ) = \bigl( ( \omega _{n}-\omega _{1} ) \circ \eta \bigr) ( t ) = ( \omega _{n}-\omega _{1} ) ( t ) \eta ( t ) =\omega _{n} ( t ) \eta ( t ) -\omega _{1} ( t ) \eta ( t ). $$

Hence, by the hypothesis of the theorem, we obtain

$$ \omega _{n}\circ \eta -\omega _{1}\circ \eta \rightarrow \omega _{1} \circ \eta -\omega _{1}\circ \eta \rightarrow 0 \quad\text{as }n \rightarrow \infty. $$

Hence, the first part of the theorem is completely proved. To prove (ii), let \(\omega _{1}\) and \(\omega _{2}\) be in \(L_{\mathbb{M}}\) and \(( \upsilon _{n} ) \in \Delta _{q,\mathbb{M}}\). If \(\omega _{1}\circ \upsilon _{n}=\omega _{2}\circ \upsilon _{n}\), then \(\omega _{1} ( t ) \upsilon _{n} ( t ) =\omega _{2} ( t ) \upsilon _{n} ( t ) \). Hence,

$$ ( \omega _{1}-\omega _{2} ) ( t ) \upsilon _{n} ( t ) =0\quad\text{for all }t\in \mathbb{R}_{q,+}. $$

Therefore, \(( \omega _{1}-\omega _{2} ) ( t ) =0\) for all \(\mathbb{R}_{q,+}\). Thus, \(\omega _{1}=\omega _{2}\) in \(L_{\mathbb{M}}\). The proof of (iii) is similar. Hence, the theorem is completely proved.

If \(( \omega _{n} ) \in L_{\mathbb{M}}\) and \(( \upsilon _{n} ) \in \Delta _{q,\mathbb{M}}\), then the pair \(( \omega _{n},\upsilon _{n} ) \) \(( \text{or }\frac{\omega _{n}}{\upsilon _{n}} ) \) is said to be a q-quotient of sequences if

$$ \omega _{n}\circ \upsilon _{m}=\omega _{m}\circ \upsilon _{n}, \quad\forall n,m\in \mathbb{N}. $$

Therefore, if \(\frac{\omega _{n}}{\epsilon _{n}}\) and \(\frac{g_{n}}{\upsilon _{n}}\) are q-quotients of sequences and \(\omega \in L_{\mathbb{M}}\), then it is easy to see that

$$ \frac{\omega \circ \epsilon _{n}}{\epsilon _{n}}\quad\text{and}\quad \frac{\omega _{n}\circ \epsilon _{n}+g_{n}\circ \epsilon _{n}}{\epsilon _{n}\circ \upsilon _{n}} $$

are q-quotients of sequences. Furthermore, it is easy to see the following equivalence relations:

$$ \frac{\omega _{n}}{\epsilon _{n}\circ \omega }\sim \frac{\omega _{n}\circ \omega }{\epsilon _{n}}\quad\text{and}\quad \frac{\omega _{n}}{\epsilon _{n}\circ g_{n}}\sim \frac{\omega _{n}\circ g_{n}}{\epsilon _{n}}. $$

Two q-quotients of sequences \(\frac{\omega _{n}}{\epsilon _{n}}\) and \(\frac{g_{n}}{\upsilon _{n}}\) are said to be equivalent if \(\omega _{n}\circ \upsilon _{m}=g_{m}\circ \epsilon _{n},\forall n,m \in \mathbb{N}\). The equivalent class \(\breve{w}= ( \frac{\omega _{n}}{\epsilon _{n}} ) \) of q-quotients of sequences containing \(\frac{\varphi _{n}}{\epsilon _{n}}\) is said to be a q-Boehmian. The space of such q-Boehmians is denoted by \(\mathbb{B}_{\mathbb{M}}\). □

Remark 16

For two q-Boehmians \(\breve{w}= ( \frac{\omega _{n}}{\epsilon _{n}} ) \) and \(\breve{z}= ( \frac{g_{n}}{\upsilon _{n}} ) \) in \(\mathbb{B}_{\mathbb{M}}\), the following are well defined on \(\mathbb{B}_{\mathbb{M}}\):

$$\begin{aligned} &\mathrm{(i)}\quad \breve{w}+\breve{z}= \biggl( \frac{\omega _{n}\circ \epsilon _{n}+g_{n}\circ \epsilon _{n}}{\epsilon _{n}\circ \upsilon _{n}} \biggr), \\ &\mathrm{(ii)}\quad \beta \breve{w}= \biggl( \frac{\beta \omega _{n}}{\epsilon _{n}} \biggr), \\ &\mathrm{(iii)}\quad \breve{w}\circ \breve{z}= \biggl( \frac{\omega _{n}\circ g_{n}}{\epsilon _{n}\circ \upsilon _{n}} \biggr), \\ &\mathrm{(iv)}\quad D^{k}\breve{w}= \biggl( \frac{D^{k}\omega _{n}}{\epsilon _{n}} \biggr), \\ &\mathrm{(v)} \quad\breve{w}\circ \omega = \biggl( \frac{\omega _{n}\circ \omega }{\epsilon _{n}} \biggr), \end{aligned}$$

where \(k\in \mathbb{R}\), \(\beta \in \mathbb{C}\) and \(D^{k}\breve{w}\) is the kth derivative of , and \(\psi \in L_{\mathbb{M}}\).

Definition 17

(i) For \(n=1,2,3,\ldots \) and \(\breve{w}_{n},\breve{w}\in \mathbb{B}_{\mathbb{M}}\), the sequence \(( \breve{w}_{n} ) \) is said to be \(\delta _{q}\)-convergent to , denoted by \(\delta _{q}-\lim_{n\rightarrow \infty }\breve{w}_{n}=\breve{w}\), provided there can be found a q-delta sequence \(( \upsilon _{n} ) \) such that

$$ ( \breve{w}_{n}\circ \upsilon _{k} ), ( \breve{w} \circ \upsilon _{k} ) \quad\text{in }L_{\mathbb{M}}\ ( \forall n,k\in \mathbb{N} ) \quad\text{and}\quad\lim_{n\rightarrow \infty } \breve{w}_{n}\circ \upsilon _{k}=\breve{w}\circ \upsilon _{k}\quad\text{in }L_{\mathbb{M}}\ ( \forall \text{ }k\in \mathbb{N} ). $$

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

$$ ( \breve{w}_{n}-\breve{w} ) \circ \upsilon _{n} \in L_{\mathbb{M}} \quad( \forall n\in \mathbb{N} ) \quad\text{and}\quad \lim_{n\rightarrow \infty } ( \breve{w}_{n}-\breve{w} ) \circ \upsilon _{n}=0 \quad\text{in }L_{\mathbb{M}}. $$

Corollary 18

(i) Let \(\omega \in L_{\mathbb{M}}\) and \(( \upsilon _{n} ) \in \Delta _{q}\) be fixed. Then the mapping

$$ \omega \rightarrow \breve{w}, $$

where \(\breve{w}=\frac{\omega \circ \upsilon _{n}}{\upsilon _{n}}\) is an injective mapping from \(L_{\mathbb{M}}\) into \(\mathbb{B}_{\mathbb{M}}\).

(ii) Let \(( \upsilon _{n} ) \in \Delta _{q,\mathbb{M}}\). Then, if \(\omega _{n}\rightarrow \omega \) in \(L_{\mathbb{M}}\) as \(n\rightarrow \infty \), then for all \(k\in \mathbb{N}\),

$$ \omega _{n}\circ \upsilon _{k}\rightarrow \omega \circ \upsilon _{k} \quad\textit{and}\quad\breve{w}_{n}\rightarrow \breve{w} \quad\textit{in }\mathbb{B}_{ \mathbb{M}}\textit{ as }n \rightarrow \infty. $$
(22)

Therefore, it can be easily checked that \(L_{\mathbb{M}}\) may be identified as a subspace of \(\mathbb{B}_{\mathbb{M}}\).

The above corollary can be stated as follows.

Corollary 19

The q-embedding \(\psi \rightarrow \breve{w}\), \(\breve{w}=\frac{\omega \circ \upsilon _{n}}{\upsilon _{n}}\), of the space \(L_{\mathbb{M}}\) into the space \(\mathbb{B}_{\mathbb{M}}\) is continuous.

The q-Mellin transform of the generalized q-theory

This section aims to discuss a definition and some basic properties of the generalized q-Mellin transform in a context of the new q-theory. All results are brief and concise, and may give the reader a general overview of the generalized q-theory of the Mellin operator. However, by virtue of the preceding analysis, we introduce the following definition.

Definition 20

Let \(\frac{g_{n}}{\varepsilon _{n}}\in \mathbb{B}\), then we define the q-Mellin transform of the q-Boehmian \(\frac{g_{n}}{\varepsilon _{n}}\) as

$$ \mathbb{M}_{q}\frac{g_{n}}{\varepsilon _{n}}=\tilde{\omega }_{n}, $$
(23)

where \(\tilde{\omega }_{n}=\frac{\omega _{n}}{\upsilon _{n}},\omega _{n}=M_{q}g\), and \(\upsilon _{n}=M_{q}\varepsilon _{n}\). Indeed \(\tilde{\omega }_{n}\) belongs to \(\mathbb{B}_{\mathbb{M}}\).

Theorem 21

The operator \(\mathbb{M}_{q}:\mathbb{B}\rightarrow \mathbb{B}_{\mathbb{M}}\) is sequentially continuous, i.e., if \(\Delta _{q}-\lim_{k\rightarrow \infty }\tilde{\omega }_{n,k}= \tilde{\omega }_{n}\) in \(\mathbb{B}\), then \(\Delta _{q,\mathbb{M}}-\lim_{n\rightarrow \infty }\mathbb{M}_{q} \tilde{\omega }_{n,k}=\mathbb{M}_{q}\tilde{\omega }_{n}\) in \(\mathbb{B}_{\mathbb{M}}\).

Proof

Let \(\Delta _{q}-\lim_{k\rightarrow \infty }\tilde{\omega }_{n,k}=\tilde{\omega }_{n}\) in \(\mathbb{B}\), then there is \((\varepsilon _{n} )\in \Delta _{q}\) such that

$$ \Delta _{q}-\lim_{n\rightarrow \infty } ( \tilde{\omega }_{n,k}- \tilde{\omega }_{n} ) \overset{q}{\bullet }\varepsilon _{n}=0 \quad\text{in }\mathbb{B}. $$

The continuity of the integral operator gives

$$ \Delta _{q,\mathbb{M}}-\lim_{n\rightarrow \infty }\mathbb{M}_{q} \bigl( ( \tilde{\omega }_{n,k}-\tilde{\omega }_{n} ) \overset{q}{\bullet }\varepsilon _{n} \bigr) =\Delta -\lim _{n\rightarrow \infty } \bigl( ( \mathbb{M}_{q}\tilde{\omega }_{n,k}-\mathbb{M}_{q} \tilde{\omega }_{n} ) \circ \upsilon _{n} \bigr) =0, $$

where \(\mathbb{M}_{q}\varepsilon _{n}=\upsilon _{n}\). Thus, we have \(\Delta _{q,\mathbb{M}}-\lim_{n\rightarrow \infty }\mathbb{M}_{q} \tilde{\omega }_{n,k}=\mathbb{M}_{q}\tilde{\omega }_{n}\) in \(\mathbb{B}_{\mathbb{M}}\).

This finishes the proof of the theorem. □

Theorem 22

(i) \(\mathbb{M}_{q}\) is a linear isomorphism from the space \(\mathbb{B}\) onto the space \(\mathbb{B}_{\mathbb{M}}\).

(ii) \(\mathbb{M}_{q}\) is continuous with respect to \(\delta _{q}\) and \(\Delta _{q}\)-convergence.

(iii) The operator \(\mathbb{M}_{q}\) coincides with the operator \(M_{q}\).

Proof

We prove Part (iii) since similar proofs for Part (i)–Part (ii) are available in literature. Let \(g\in L_{q}^{1} ( \mathbb{R}_{q,+} ) \) and \(\frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}}\) be its representative in \(\mathbb{B}\), where \(( \varepsilon _{n} ) \in \Delta _{q}\) \(( \forall n\in \mathbb{N} ) \). Clearly, for all \(n\in \mathbb{N}{\small,}\) \(( \varepsilon _{n} ) \) is independent from the representative. Let \(\mathbb{M}_{q}\varepsilon _{n}=\upsilon _{n}\), then, by the q-convolution theorem, we get

$$ \mathbb{M}_{q} \frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}}=\mathbb{M}_{q} \frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}}=\frac{M_{q}g\circ M_{q}\varepsilon _{n}}{M_{q}\varepsilon _{n}}=M_{q}g\circ \frac{M_{q}\varepsilon _{n}}{M_{q}\varepsilon _{n}}= \omega \circ \frac{\upsilon _{n}}{\upsilon _{n}}. $$

Hence, the q-Boehmian \(\frac{\omega \circ \upsilon _{n}}{\upsilon _{n}}\) is the representative of \(\mathbb{M}_{q}\) in the space \(L_{\mathbb{M}}\), where \(\omega =M_{q}g\).

The proof is, therefore, ended. □

We introduce the inverse transform of \(\mathbb{M}_{q}\) as follows.

Definition 23

We define the inverse integral operator of \(\mathbb{M}_{q}\) of a q-Boehmian \(\frac{\omega _{n}}{\upsilon _{n}}\) in \(\mathbb{B}_{\mathbb{M}}\) as follows:

$$ \mathbb{N}_{q}\frac{\omega _{n}}{\upsilon _{n}}= \frac{g_{n}}{\varepsilon _{n}}\in \mathbb{B}, $$

where \(\upsilon _{n}=\mathbb{M}_{q}\varepsilon _{n}\) and \(\omega _{n}=M_{q}g_{n}\) for some \(( \varepsilon _{n} ) \in \Delta _{q}\) and \(\{g_{n}\}\in L_{q}^{1} ( \mathbb{R}_{q,+} ) \).

Theorem 24

Let \(\frac{\omega _{n}}{\upsilon _{n}}\in \mathbb{B}_{\mathbb{M}}\) and \(\omega \in L_{\mathbb{M}}\). Then we have

$$ \mathbb{N}_{q} \biggl( \frac{\omega _{n}}{\upsilon _{n}}\circ \omega \biggr) =\frac{g_{n}}{\varepsilon _{n}}\overset{q}{\bullet }g\quad\textit{and}\quad \mathbb{M}_{q} \biggl( \frac{g_{n}}{\varepsilon _{n}}\overset{q}{\bullet }g \biggr) =\frac{\omega _{n}}{\upsilon _{n}}\circ \omega. $$

Proof

Assume \(\frac{\omega _{n}}{\upsilon _{n}}\in \mathbb{B}_{\mathbb{M}}\) where \(\omega _{n}=M_{q}g_{n}\). Then, for every \(\omega =M_{q}g\in L_{\mathbb{M}}\) and \(\upsilon _{n}=M_{q}\varepsilon _{n}\), we have

$$ \mathbb{N}_{q} \biggl( \frac{\omega _{n}}{\upsilon _{n}}\circ M_{q}g \biggr) =\mathbb{N}_{q}\frac{\omega _{n}\circ \omega }{\upsilon _{n}}= \mathbb{N}_{q}\frac{M_{q} ( g_{n}\overset{q}{\bullet }g ) }{\upsilon _{n}}=\frac{g_{n}\overset{q}{\bullet }g}{\varepsilon _{n}}= \frac{g_{n}}{\varepsilon _{n}}\overset{q}{\bullet }g. $$

The proof of the first part is finished. The proof of the second part is almost similar. Hence, we omit the details.

This completely ends the proof of the theorem. □

Conclusion

This paper has given an extension of the quantum theory of the q-Mellin transform operator [40] to sets of q-generalized functions named q-Boehmians and q-ultraBoehmians. Every element g in the function space \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) is identified as a member in the generalized space \(\mathbb{B}\) by the identification formula

$$ g\rightarrow \frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}}, $$

where \(( \varepsilon _{n} ) \) is an arbitrary delta sequence. It also shows that the q-embedding

$$ g\rightarrow \breve{w},\text{ }\breve{w}= \frac{g\overset{q}{\bullet }\varepsilon _{n}}{\varepsilon _{n}} $$

of the space \(L_{q}^{1} ( \mathbb{R}_{q,+} ) \) into the space \(\mathbb{B}\) is continuous, \(( \varepsilon _{n} ) \) being an arbitrary q-delta sequence. The q-Mellin transform operator is extended to the generalized q-calculus theory, and many properties are discussed. Further, the inversion of the q-Mellin transform operator is also discussed.

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The author would like to express deepest thanks to the reviewers for their insightful comments on his paper.

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Al-Omari, S. Estimates and properties of certain q-Mellin transform on generalized q-calculus theory. Adv Differ Equ 2021, 233 (2021). https://doi.org/10.1186/s13662-021-03391-z

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MSC

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

Keywords

  • q-delta sequences
  • q-Mellin
  • q-convolution
  • q-calculus
  • q-Boehmian
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