scholarly journals The q-Sumudu transform and its certain properties in a generalized q-calculus theory

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shrideh Khalaf Al-Omari
Keyword(s):  
Inversion Formula ◽  
Generalized Functions ◽  
Convolution Theorem ◽  
General Properties ◽  
Integrable Functions ◽  
Sumudu Transform ◽  
Convolution Products ◽  
Set Up

AbstractIn this paper we consider a generalization to the q-calculus theory in the space of q-integrable functions. We introduce q-delta sequences and develop q-convolution products to derive certain q-convolution theorem. By using the concept of q-delta sequences, we establish various axioms and set up q-spaces of generalized functions named q-Boehmian spaces. The new assigned spaces of q-generalized functions are acceptable and compatible with the classical spaces of the ordinary functions. Consequently, we extend the generalized q-Sumudu transform to the sets of q-Boehmian spaces. On top of that, we nominate the canonical q-embeddings between the q-integrable sets of functions and the q-integrable sets of q-Boehmians. Furthermore, we address the general properties of the generalized q-Sumudu transform and its inversion formula in some detail.

scholarly journals δ-β-Gabor integral operators for a space of locally integrable generalized functions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shrideh Khalaf Al-Omari ◽  
Dumitru Baleanu ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Integral Operator ◽  
Inversion Formula ◽  
Integral Operators ◽  
Generalized Functions ◽  
Convolution Theorem ◽  
Consistency Results ◽  
Convolution Products ◽  
One To One ◽  
The Given

Abstract In this article, we give a definition and discuss several properties of the δ-β-Gabor integral operator in a class of locally integrable Boehmians. We derive delta sequences, convolution products and establish a convolution theorem for the given δ-β-integral. By treating the delta sequences, we derive the necessary axioms to elevate the δ-β-Gabor integrable spaces of Boehmians. The said generalized δ-β-Gabor integral is, therefore, considered as a one-to-one and onto mapping continuous with respect to the usual convergence of the demonstrated spaces. In addition to certain obtained inversion formula, some consistency results are also given.

scholarly journals Certain fundamental properties of generalized natural transform in generalized spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shrideh Khalaf Al-Omari ◽  
Serkan Araci
Keyword(s):  
Inversion Formula ◽  
Acta Math ◽  
Generalized Functions ◽  
General Nature ◽  
Qualitative Behavior ◽  
Fundamental Properties ◽  
Integrable Functions ◽  
Extended Operator ◽  
Definition Of ◽  
Space Of Integrable Functions

AbstractThis paper considers the definition and the properties of the generalized natural transform on sets of generalized functions. Convolution products, convolution theorems, and spaces of Boehmians are described in a form of auxiliary results. The constructed spaces of Boehmians are achieved and fulfilled by pursuing a deep analysis on a set of delta sequences and axioms which have mitigated the construction of the generalized spaces. Such results are exploited in emphasizing the virtual definition of the generalized natural transform on the addressed sets of Boehmians. The constructed spaces, inspired from their general nature, generalize the space of integrable functions of Srivastava et al. (Acta Math. Sci. 35B:1386–1400, 2015) and, subsequently, the extended operator with its good qualitative behavior generalizes the classical natural transform. Various continuous embeddings of potential interests are introduced and discussed between the space of integrable functions and the space of integrable Boehmians. On another aspect as well, several characteristics of the extended operator and its inversion formula are discussed.

scholarly journals A new aspect of generalized integral operator and an estimation in a generalized function theory

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shrideh Al-Omari ◽  
Hassan Almusawa ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Integral Operator ◽  
Inversion Formula ◽  
Function Theory ◽  
Generalized Functions ◽  
Generalized Function ◽  
Dunkl Operator ◽  
Generalized Function Theory ◽  
Convolution Products

AbstractIn this paper we investigate certain integral operator involving Jacobi–Dunkl functions in a class of generalized functions. We utilize convolution products, approximating identities, and several axioms to allocate the desired spaces of generalized functions. The existing theory of the Jacobi–Dunkl integral operator (Ben Salem and Ahmed Salem in Ramanujan J. 12(3):359–378, 2006) is extended and applied to a new addressed set of Boehmians. Various embeddings and characteristics of the extended Jacobi–Dunkl operator are discussed. An inversion formula and certain convergence with respect to δ and Δ convergences are also introduced.

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

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shrideh Al-Omari
Keyword(s):  
Inversion Formula ◽  
Mellin Transform ◽  
Generalized Functions ◽  
Convolution Theorem ◽  
Equivalence Relations

AbstractThis 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.

scholarly journals A study on a class of modified Bessel-type integrals in a Fréchet space of Boehmians

2019 ◽  
Vol 38 (4) ◽  
pp. 145-156 ◽  
Author(s):  
Shrideh Khalaf Al-Omari
Keyword(s):  
Generalized Functions ◽  
Convolution Theorem ◽  
Fréchet Space ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Basic Properties ◽  
Type Integral ◽  
Classical Integral ◽  
Convolution Products

In this paper, an attempt is being made to discuss a class of modified Bessel- type integrals on a set of generalized functions known as Boehmians. We show that the modified Bessel-type integral, with appropriately defined convolution products, obeys a fundamental convolution theorem which consequently justifis pursuing analysis in the Boehmian spaces. We describe two Fréchet spaces of Boehmians and extend the modifid Bessel-type integral between the diferent spaces. Furthermore, a convolution theorem and a class of basic properties of the extended integral such as linearity, continuity and compatibility with the classical integral, which provide a convenient extention to the classical results, have been derived

scholarly journals A generalized inversion formula for the continuous Jacobi transform

1987 ◽  
Vol 10 (4) ◽  
pp. 671-692 ◽  
Author(s):  
Ahmed I. Zayed
Keyword(s):  
Analytic Function ◽  
Inversion Formula ◽  
Generalized Functions ◽  
Generalized Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Inversion ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Jacobi Transform

In this paper we extend the definition of the continuous Jacobi transform to a class of generalized functions and obtain a generalized inversion formula for it. As a by-product of our technique we obtain a necessary and sufficient condition for an analytic functionF(λ)inReλ>0to be the continuous Jacobi transform of a generalized function.

scholarly journals A New Version of the Generalized Krätzel-Fox Integral Operators

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 222 ◽  
Author(s):  
Shrideh Al-Omari ◽  
Ghalib Jumah ◽  
Jafar Al-Omari ◽  
Deepali Saxena
Keyword(s):  
Integral Operator ◽  
Integral Operators ◽  
Convolution Theorem ◽  
Fréchet Space ◽  
Frechet Space ◽  
Generalized Convolution ◽  
Real Numbers ◽  
Convolution Products ◽  
Fox’S H Function ◽  
Krätzel Integral

This article deals with some variants of Krätzel integral operators involving Fox’s H-function and their extension to classes of distributions and spaces of Boehmians. For real numbers a and b > 0 , the Fréchet space H a , b of testing functions has been identified as a subspace of certain Boehmian spaces. To establish the Boehmian spaces, two convolution products and some related axioms are established. The generalized variant of the cited Krätzel-Fox integral operator is well defined and is the operator between the Boehmian spaces. A generalized convolution theorem has also been given.

scholarly journals On Katugampola Fourier Transform

2019 ◽  
Vol 2019 ◽  
pp. 1-6 ◽  
Author(s):  
Tariq O. Salim ◽  
Atta A. K. Abu Hany ◽  
Mohammed S. El-Khatib
Keyword(s):  
Fourier Transform ◽  
Inversion Formula ◽  
Convolution Theorem ◽  
The Fourier Transform

The aim of this article is to introduce a new definition for the Fourier transform. This new definition will be considered as one of the generalizations of the usual (classical) Fourier transform. We employ the new Katugampola derivative to obtain some properties of the Katugampola Fourier transform and find the relation between the Katugampola Fourier transform and the usual Fourier transform. The inversion formula and the convolution theorem for the Katugampola Fourier transform are considered.

scholarly journals Two index laws for fractional integrals and derivatives

1972 ◽  
Vol 14 (4) ◽  
pp. 385-410 ◽  
Author(s):  
E. R. Love
Keyword(s):  
Integral Equations ◽  
Fractional Derivatives ◽  
Addition Theorem ◽  
Positive Real Part ◽  
Generalized Functions ◽  
Fractional Integrals ◽  
Positive Real ◽  
Integrable Functions ◽  
Hypergeometric Integral ◽  
Fractional Integrals And Derivatives

SummaryThe first index law, or addition theorem, is well known. The second is much less well known; but both have been found to be of importance in recent studies of hypergeometric integral equations. The first law has usually been considered only in the simple case of orders of integration which have positive real part, or in the context of generalized functions. Arising out of the need to manipulate expressions involving several fractional integrals and derivatives, our aim here is to establish both laws for all combinations of complex orders of integration and differentiation, and for nearly all functions for which the fractional derivatives involved exist as locally integrable functions.

scholarly journals Complex inversion formula for the distributional Stieltjes transform

1987 ◽  
Vol 30 (3) ◽  
pp. 363-371 ◽  
Author(s):  
Stevan Pilipović
Keyword(s):  
Inversion Formula ◽  
Generalized Functions ◽  
Schwartz Space ◽  
Stieltjes Transform ◽  
Definition Of

There are several approaches to the Stieltjes transform of generalized functions ([1, 10, 5, 6, 3, 2]). In this paper we use the definition of the distributional Stieltjes transform of index ρ (ρ ∈ ℝ\(−ℕ0); ℕ0 = ℕ∪{0}), Sρ-transform, given by Lavoine and Misra [3]. The Sρ-transform is defined for a subspace of the Schwartz space (ℝ) while in [10, 5, 6, 2] the Stieltjes transform is defined for the elements of appropriate spaces of generalized functions. In these spaces differentiation is not defined which means that the Stieltjes transform of some important distributions, for example δ(k)(x − a), a≧0, k ∈ ℕ, is meaningless in the sense of [10, 5, 6, 2]. It is easy to see that the distributions δ(k)(x − a), a≧0, k ∈ ℕ, have the Sρ-transform for ρ>−k, ρ∈ℝ\(−ℕ0). These facts favour the approach to the Stieltjes transform given in [3].

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