Space Of Boehmians Research Articles

Mellin transform on compactly supported Boehmians

We extend the Mellin transform on a new Boehmian space by using the compactly supported distributions and where a Mellin-type convolution becomes the product for this space.

Offset Linear Canonical Stockwell Transform for Boehmians

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, it is one-to-one, linear, and continuous with respect to 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.

On the Generalized Lambert Transform over Lebesgue and Boehmian Spaces

On the Generalized Lambert Transform over Lebesgue and Boehmian Spaces

Hermite transform for distribution and Boehmian space

Hermite transform involves weigth function and Hermite polynomial as its kernel is discussed. The Hermite transform and its basic properties are extended to the distribution spaces and to the space of integrable Boehmian

On Stieltjes type transforms over Boehmian spaces

We extend integral transforms of Stieltjes type to a larger space of Boehmians than that used in previous studies. Our analysis includes the Stieltjes transform, the Poisson transform (Widder’s potential transform) and the generalized Stieltjes transform.

The Fractional Hilbert Transform of Generalized Functions

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.

On Distributional Sadik Transform and Boehmian Space

In present article the author has tried to define Sadik Transform in distributional sense. Some of the properties of Sadik transform are proved. Boehmian Space for Sadik distributional transform is defined along with some basic properties.

Fractional Hartley transform on $G$-Boehmian space

Using a special type of fractional convolution, a $G$-Boehmian space $\mathcal{B}_\alpha$ containing integrable functions on $\mathbb{R}$ is constructed. The fractional Hartley transform ({\sc frht}) is defined as a linear, continuous injection from $\mathcal{B}_\alpha$ into the space of all continuous functions on $\mathbb{R}$. This extension simultaneously generalizes the fractional Hartley transform on $L^1(\mathbb{R})$ as well as Hartley transform on an integrable Boehmian space.

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

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.

Certain fundamental properties of generalized natural transform in generalized spaces

This 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.

The Lambert transform over distributions of compact support, $$L^1$$-functions and Boehmian spaces

In this paper, we study the Lambert transform over distributions of compact support on $$(0,\infty )$$ . We obtain an inversion formula for this transform and we prove a Parseval-type relation for the Lambert transform of functions in $$L^1 ((0,\infty ))$$ . We also extend this transform to Boehmian spaces.

Quaternionic Fractional Fourier Transform for Boehmians

UDC 517.9 We construct a Boehmian space of quaternion valued functions using the quaternionic fractional convolution. Applying the convolution theorem, the quaternionic fractional Fourier transform is extended to the context of Boehmians and its properties are established.

The Watson Fourier transform on a certain class of generalized functions

In this paper, we first build an appropriate Boehmian space on which a classical theory of the Watson Fourier transform is extended. We provide convolution products, convolution theorems and generate their associated spaces of Boehmian. The properties of the extended Watson Fourier transform are also established. Moreover, continuity with respect to $$\delta$$ and $$\Delta$$ -convergence is discussed.

\u03b4-\u03b2-Gabor integral operators for a space of locally integrable generalized functions

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.

Extension of generalized Fox\u2019s H-function operator to certain set of generalized integrable functions

In this article, we investigate the so-called Inayat integral operator T_{p,q}^{m,n}, p,q,m,nin mathbb{Z}, 1leq mleq q, 0leq nleq p , on classes of generalized integrable functions. We make use of the Mellin-type convolution product and produce a concurrent convolution product, which, taken together, establishes the Inayat integral convolution theorem. The Inayat convolution theorem and a class of delta sequences were derived and employed for constructing sequence spaces of Boehmians. Moreover, by the aid of the concept of quotients of sequences, we present a generalization of the Inayat integral operator in the context of Boehmians. Various results related to the generalized integral operator and its inversion formula are also derived.

Estimates of certain paraxial diffraction integral operator and its generalized properties

This paper aims to discuss a generalization of certain paraxial diffraction integral operator in a class of generalized functions. At the start of this paper, we propose a convolution formula and establish certain convolution theorem. Then, with the addition to the convolution theorem, we consider a set of approximating identities and substantially employ our results in generating sets of integrable and locally integrable Boehmians. The said generalized integral operator is tested and declared to be one-to-one and onto mapping. Continuity of the generalized operator with respect to the convergence of the Boehmian spaces is obtained. Over and above, an inversion formula and consistency results are also counted.

Quaternionic fractional Fourier transform for Boehmians

UDC 517.9 We construct a Boehmian space of quaternion valued functions using the quaternionic fractional convolution. Applying the convolution theorem, the quaternionic fractional Fourier transform is extended to the context of Boehmians and its properties are established.

KAMAL TRANSFORM OF STRONG BOEHMIANS

The concept of Boehmian was motivated by the so called regular operators introduced by T.K.Boehme. The construction of Boehmians is similar to the construction of field of quotients. Several integral transforms have been extended to various class of Boehmians. We study here Kamal transform and extend it to Strong Boehmian space. This Kamal tranform is 1-1 and continuous in the space of Boehmians. Inverse Kamal transform is also defined.

An extension of the Bessel\u2013Wright 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.

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

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