Space Of Boehmians Research Articles

Stockwell transform for Boehmians

In this article, we state and prove a convolution theorem for the Stockwell transform on L 2(ℝ) and characterize the range of the transform. Applying the convolution theorem, we construct a Boehmian space which contains the Stockwell transforms of all square-integrable Boehmians. We prove that the extended Stockwell transform on square-integrable Boehmians is consistent with the classical Stockwell transform on square-integrable functions and is linear, one to one, continuous with respect to δ-convergence as well as Δ-convergence. We also characterize the range of the extended Stockwell transform on the space of square-integrable Boehmians.

An estimate of Sumudu transforms for Boehmians

The space of Boehmians is constructed using an algebraic approach that utilizes convolution and approximate identities or delta sequences. A proper subspace can be identified with the space of distributions. In this paper, we first construct a suitable Boehmian space on which the Sumudu transform can be defined and the function space S can be embedded. In addition to this, our definition extends the Sumudu transform to more general spaces and the definition remains consistent for S elements. We also discuss the operational properties of the Sumudu transform on Boehmians and finally end with certain theorems for continuity conditions of the extended Sumudu transform and its inverse with respect to δ- and Δ-convergence.

FOURIER SINE AND COSINE TRANSFORMS ON BOEHMIAN SPACES

We construct suitable Boehmian spaces which are properly larger than [Formula: see text] and we extend the Fourier sine transform and the Fourier cosine transform more than one way. We prove that the extended Fourier sine and cosine transforms have expected properties like linear, continuous, one-to-one and onto from one Boehmian space onto another Boehmian space. We also establish that the well known connection among the Fourier transform, Fourier sine transform and Fourier cosine transform in the context of Boehmians. Finally, we compare the relations among the different Boehmian spaces discussed in this paper.

Fractional Fourier Transform of Boehmians

In this paper we introduce two spaces of Boehmians each of which contains the dual of a certain space of entire functions. Both these spaces of Boehmians are shown to be isomorphic to each other under the Fractional Fourier transform. We extend the theory of the Fractional Fourier transform on this new Boehmians space. Mathematics Subject Classification: (2000) 44A15, 44A40, 46F12, 44-99.

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.

Unified Treatment of the Krätzel Transformation for Generalized Functions

We discuss a generalization of the Krätzel transforms on certain spaces of ultradistributions. We have proved that the Krätzel transform of an ultradifferentiable function is an ultradifferentiable function and satisfies its Parseval's inequality. We also provide a complete reading of the transform constructing two desired spaces of Boehmians. Some other properties of convergence and continuity conditions and its inverse are also discussed in some detail.

Note on Boehmians for Class of Optical Fresnel Wavelet Transforms

We extend the Fresnel-wavelet transform to the context of generalized functions, namely, Boehmians. At first, we study the Fresnel-wavelet transform in the sense of distributions of compact support. Based on this concept, we introduce two new spaces of Boehmians and proving certain related results. Further, we show that the extended transform establishes a linear and an isomorphic mapping between the Boehmian spaces. Moreover, conditions of continuity of the extended transform and its inverse with respect toδand Δ convergence are discussed in some details.

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.

Gelfand transform for a Boehmian space of analytic functions

Gelfand transform for a Boehmian space of analytic functions

S-asymptotic properties of Boehmians

The S-asymptotic behaviour for the space of Boehmians in d-dimensions is defined and studied. An application for elliptic partial differential equations is given.

Poisson transform on Boehmians

Convolution theorem for Poisson transform on compactly supported distributions is obtained. Applying the convolution theorem, Poisson transform is extended as a linear continuous map from a suitable Boehmian space into another Boehmian space satisfying the convolution theorem.

One point compactification for generalized quotient spaces

The concept of Generalized function spaces which were introduced and studied by Zemanian are further generalized as Boehmian spaces or as generalized quotient spaces in the recent literature. Their topological structure, notions of convergence in these space sare also investigated. Some sufficient conditions for the metrizability are also obtained. In this paper we shall assume that a generalized quotient space is non-compact and realize its one point compactification as a quotient space.

The Laplace transform on a Boehmian space

The Laplace transform on a Boehmian space

Boehmians of type S and their Fourier transforms

Function spaces of type S are introduced and investigated in the literature. They are also applied to study the Cauchy problem. In this paper we shall extend the concept of these spaces to the context of Boehmian spaces and study the Fourier transform theory on these spaces. These spaces enable us to combine the theory of Fourier transform on these function spaces as well as their dual spaces.

Boehmians of type S and their Fourier transforms

Function spaces of type S are introduced and investigated in the literature. They are also applied to study the Cauchy problem. In this paper we shall extend the concept of these spaces to the context of Boehmian spaces and study the Fourier transform theory on these spaces. These spaces enable us to combine the theory of Fourier transform on these function spaces as well as their dual spaces.

Kontorovich–Lebedev transform for Boehmians

In this paper, we define and study a continuous linear map from one space of Boehmians into another for the Kontorovich–Lebedev transform. The Kontorovich–Lebedev transform is further investigated for compactly supported distributions.

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.

CONVOLUTION THEOREMS FOR WAVELET TRANSFORM ON TEMPERED DISTRIBUTIONS AND THEIR EXTENSION TO TEMPERED BOEHMIANS

We define a new convolution ⊗ : 𝒮'(ℝ × ℝ+) × 𝒟(ℝ) → 𝒮'(ℝ × ℝ+) and derive the convolution theorems for wavelet transform and dual wavelet transform in the context of tempered distributions. By using the new convolution, we construct a Boehmian space containing the tempered distributions on ℝ × ℝ+. Applying the convolution theorems in the context of tempered distributions, we also extend the wavelet transform and dual wavelet transform between the tempered Boehmian space and the new Boehmian space as linear continuous maps with respect to δ-convergence and Δ-convergence, satisfying the convolution theorems.

Fourier sine (cosine) transform for ultradistributions and their extensions to tempered and ultraBoehmian spaces

This paper deals with ultradistributions for the Fourier sine (cosine) transform on a certain dual testing function space and tempered Boehmian and ultraBoehmian spaces.