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16 pages, 310 KiB

Open AccessArticle

On (p,q)-Analogs of the α-th Fractional Fourier Transform and Some (p,q)-Generalized Spaces

by Shrideh Al-Omari and Wael Salameh

Symmetry 2024, 16(10), 1307; https://doi.org/10.3390/sym16101307 - 3 Oct 2024

Viewed by 1416

In this article, the

(p, q)

-analogs of the

α

-th fractional Fourier transform are provided, along with a discussion of their characteristics in specific classes of

(p, q)

-generalized functions. Two spaces of infinitely [...] Read more.

In this article, the

(p, q)

-analogs of the

α

-th fractional Fourier transform are provided, along with a discussion of their characteristics in specific classes of

(p, q)

-generalized functions. Two spaces of infinitely

(p, q)

-differentiable functions are defined by introducing two

(p, q)

-differential symmetric operators. The

(p, q)

-analogs of the

α

-th fractional Fourier transform are demonstrated to be continuous and linear between the spaces under discussion. Next, theorems pertaining to specific convolutions are established. This leads to the establishment of multiple symmetric identities, which in turn requires the construction of

(p, q)

-generalized spaces known as

(p, q)

-Boehmians. Finally, in addition to deriving the inversion formulas, the generalized

(p, q)

- analogs of the

α

-th fractional Fourier transform are introduced, and their general properties are discussed. Full article

(This article belongs to the Special Issue Special Functions, Integral Transforms and Polynomial Sequences in Real World with Symmetry)

12 pages, 264 KiB

Open AccessArticle

Notes on q-Gamma Operators and Their Extension to Classes of Generalized Distributions

by Shrideh Al-Omari, Wael Salameh and Sharifah Alhazmi

Symmetry 2024, 16(10), 1294; https://doi.org/10.3390/sym16101294 - 2 Oct 2024

Viewed by 862

Abstract

This paper discusses definitions and properties of q-analogues of the gamma integral operator and its extension to classes of generalized distributions. It introduces q-convolution products, symmetric q-delta sequences and q-quotients of sequences, and establishes certain convolution theorems. The convolution theorems are utilized to accomplish q-equivalence classes of generalized distributions called q-Boehmians. Consequently, the q-gamma operators are therefore extended to the generalized spaces and performed to coincide with the classical integral operator. Further, the generalized q-gamma integral is shown to be linear, sequentially continuous and continuous with respect to some involved convergence equipped with the generalized spaces. Full article

(This article belongs to the Special Issue Discussion of Properties and Applications of Integral Transform)

18 pages, 335 KiB

Open AccessArticle

Offset Linear Canonical Stockwell Transform for Boehmians

by Navneet Kaur, Bivek Gupta, Amit K. Verma and Ravi P. Agarwal

Mathematics 2024, 12(15), 2379; https://doi.org/10.3390/math12152379 - 31 Jul 2024

Cited by 5 | Viewed by 1057

Abstract

In this article, we construct a Boehmian space using the convolution theorem that contains the offset linear canonical Stockwell transforms (OLCST) of all square-integrable Boehmians. It is also proven that the extended OLCST on square-integrable Boehmians is consistent with the traditional OLCST. Furthermore, [...] Read more.

Δ

-convergence as well as

Δ

-convergence. Subsequently, we introduce a discrete variant of the OLCST. Ultimately, we broaden the application of the presented work by examining the OLCST within the domain of almost periodic functions. Full article

(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications III)

11 pages, 255 KiB

Open AccessArticle

The Fractional Hilbert Transform of Generalized Functions

by Naheed Abdullah and Saleem Iqbal

Symmetry 2022, 14(10), 2096; https://doi.org/10.3390/sym14102096 - 8 Oct 2022

Viewed by 1806

Abstract

The fractional Hilbert transform, a generalization of the Hilbert transform, has been extensively studied in the literature because of its widespread application in optics, engineering, and signal processing. In the present work, we expand the fractional Hilbert transform that displays an odd symmetry to a space of generalized functions known as Boehmians. Moreover, we introduce a new fractional convolutional operator for the fractional Hilbert transform to prove a convolutional theorem similar to the classical Hilbert transform, and also to extend the fractional Hilbert transform to Boehmians. We also produce a suitable Boehmian space on which the fractional Hilbert transform exists. Further, we investigate the convergence of the fractional Hilbert transform for the class of Boehmians and discuss the continuity of the extended fractional Hilbert transform. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)

8 pages, 245 KiB

Open AccessArticle

A New Version of the Generalized Krätzel–Fox Integral Operators

by Shrideh K. Q. Al-Omari, Ghalib Jumah, Jafar Al-Omari and Deepali Saxena

Mathematics 2018, 6(11), 222; https://doi.org/10.3390/math6110222 - 28 Oct 2018

Cited by 8 | Viewed by 2915

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Special Functions and Applications)

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