Integrable Boehmians Research Articles

Generalized Lambert-type transforms over integrable Boehmians

We study inversion formulae and continuity properties for generalized Lambert-type transforms. We also extend those transforms over integrable Boehmians.

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.

Kamal transform for integrable Boehmians

Kamal transform for integrable Boehmians

\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.

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.

On Double Natural Transform of Boehmians

In this paper we are going to define the Double Natural transform of Bounded support and we develop this theory with the help of Zemanian technique. Here we are going to define the Integrable Boehmian and the main focus or objective of this paper is to define Double Natural transform for Integrable Boehmian, and this theory is developed and defined with the help of Mikusinski’s theory. D.Nemzer, P. Mikusinski and other discovered several applications of Boehmians. Also we establish some properties of Double Natural transform with other integral transforms. The Double Natural transform have been extended to generalized function and some of its distributional properties are obtained. Further we extend this class to Integrable Boehmians.

$Z$-transform for integrable Boehmians

Boehmians, being the specific class of the generalized functions (the theory of distributions), are exposed to applications within mathematics and beyond. This paper is involved into the investigation of the $Z$ - transform for integrable Boehmians.

On some variant of a whittaker integral operator and its representative in a class of square integrable Boehmians

This paper investigates some variant of Whittaker integral operators on a class of square integrable Boehmians. We define convolution products and derive the convolution theorem which substantially satisfy the axioms necessary for generating the Whittaker spaces of Boehmians. Relied on this analysis, we give a definition and properties of the Whittaker integral operator in the class of square integrable Boehmians. The extended Whittaker integral operator, is well-defined, linear and coincides with the classical integral in certain properties.

Cauchy representation of fractional Fourier transform for Boehmians

Results relating to fractional Fourier transform and their properties in the Lizorkin space are employed in this paper to investigate the Cauchy representation of fractional Fourier transform for integrable Boehmians. An inversion formula for the fractional Fourier transform is addressed. The conclusion remark of the paper spells the initiation for the present investigation.

Fractional Integrals of Fractional Fourier Transform for Integrable Boehmians

With an abridged introductory note on the fractional Fourier transform, this paper attempts to study the same for integrable Boehmians with regard to fractional integrals. Some relevant properties are also established.

On a Widder potential transform and its extension to a space of locally integrable Boehmians

In this paper we investigate a Widder potential transform on certain spaces of Boehmians. We construct two spaces of Boehmians. One space of Boehmians is obtained by a well-known Mellin-type convolution product. The second space is obtained by another mapping acting with the first convolution. The extended Widder potential transform is therefore a mapping, that is, well-defined, linear, continuous, with respect to δ and Δ convergence, and consistent with the classical transform. Certain theorem is also established.

Some Remarks on the Extended Hartley-Hilbert and Fourier-Hilbert Transforms of Boehmians

We obtain generalizations of Hartley-Hilbert and Fourier-Hilbert transforms on classes of distributions having compact support. Furthermore, we also study extension to certain space of Lebesgue integrable Boehmians. New characterizing theorems are also established in an adequate performance.

EXTENDED RIDGELET TRANSFORM ON DISTRIBUTIONS AND BOEHMIANS

The ridgelet transform is extended to the space of Schwartz distributions and to the space of C∞-Boehmians consistent with the classical Ridgelet transform on the space of square integrable Boehmians. The properties of the ridgelet transform like linearity, injectivity, surjectivity and continuity with respect to two notions of convergence are established. It is also justified that this extended ridgelet transform on C∞-Boehmians properly generalizes the earlier extensions of the ridgelet transform.

Hartley transform for integrable Boehmians

In the present article, we studied the properties of the Hartley transform for integrable Boehmian and proved its inversion.

Fourier transform on integrable Boehmians

The theory of Fourier transform on spaces of Boehmians is well known. Several spaces of Boehmians are constructed, and the Fourier transform on these spaces are defined and investigated in the recent literature. Most recently, Dennis Nemzer introduced a new space of Boehmians called the space of integrable Boehmians, which is properly larger than the space of all integrable functions on (the real axis) and investigates the theory of Fourier transform on this space. In this article, we shall find a necessary and sufficient condition for a continuous function to be the Fourier transform of some integrable Boehmian. This answers a question raised by Nemzer.

RIDGELET TRANSFORM ON SQUARE INTEGRABLE BOEHMIANS

The ridgelet transform is extended to the space of square integrable Boehmians. It is proved that the extended ridgelet transform R is consistent with the classical ridgelet transform R, linear, one-to-one, onto and both R,R i1 are continuous with respect to --convergence as well as ¢-convergence.

Integrable Boehmians, Fourier transforms, and Poisson’s summation formula

Boehmians are classes of generalized functions whose construction is algebraic. The first construction appeared in a paper that was published in 1981 [6]. In [8], P. Mikusinski constructs a space of Boehmians, βL1(R), in which each element has a Fourier transform. Mikusinski shows that the Fourier transform of a Boehmian satisfies some basic properties, and he also proves an inversion theorem. However, the range of the Fourier transform is not investigated. Also, Mikusinski states that βL1(R) contains some elements which are not Schwartz distributions, but no examples are given. We will address these problems in this paper. In this note, we will construct a space of Boehmians β`(R). The space of integrable functions on the real line can be identified with a proper subspace of β`(R). Each element of β`(R) has a Fourier transform which is a continuous function and satisfies a growth condition at infinity. Conditions are given which ensure that a given function is the Fourier transform of an element of β`(R).

Wavelet transforms for integrable Boehmians

By the application of continuous wavelet transforms (or windowed Fourier transform) and employing Burzyk's conjecture, the wavelet transform for integrable Boehmians is obtained. Inversion formula is also established.

The hankel transform of integrable boehmians

In this paper we investigate the Hankel transform of integrable Boehmians. The inversion theorem and some operational properties are established. Finally, the relations between the integrable Boehmians and the distributions are studied.