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This paper explores Abelian theorems associated with the real Weierstrass transform over distributions of compact support. This study contributes to both mathematical analysis and distribution theory by offering new insights into the interaction between integral transforms and compactly supported distributions. [ABSTRACT FROM AUTHOR]

Our goal in this article is to derive Abelian theorems for the two-sided Laplace transform, Mellin transform, one-sided real Laplace transform and Stieltjes transform over distributions of compact support and over certain function spaces of generalized functions. [ABSTRACT FROM AUTHOR]

In this paper, we establish new properties for the Fourier transform over the space of distributions lfk introduced by Horvath. We prove Abelian theorems for the Fourier transform over the space lfk, k 2 Z, k < 0. Continuity properties and some results concerning regular distributions are studied. We also prove that the Fourier transform is an injection from lfk, k 2 Z, k < 0, into O where this space denotes the union of the varies in Z, which have been given by Horvath. The convolution over for certain regular distributions and its relation with the usual convolution product of functions is exhibited. Finally, some illustrative examples are considered. [ABSTRACT FROM AUTHOR]

In this paper, initial and final value Abelian theorems for fractional wavelet transform of function and tempered distributions are obtained. Using Mexican hat wavelet function, an application for Abelian theorems is investigated. [ABSTRACT FROM AUTHOR]

Spaces of generalized functions known as Boehmians contain all Schwartz distributions as well as all ultradistributions of Beurling and Roumieu type. Even though these spaces are quite large, it is still possible to investigate local behavior. In this note, ultradifferential operators are introduced in the context of Boehmians. As an application, some Abelian type theorems are established for the unilateral Laplace transform. [ABSTRACT FROM AUTHOR]

Abelian theorems for complex-valued Stieltjes transforms of functions and generalized functions are obtained, which generalize all previous related results, and new results are obtained. Under limit assumptions on the function or the generalized function, the Abelian theorem results are obtained as the complex variable of the transform approaches zero or infinity within an arbitrary wedge, and in some cases arbitrarily, in the right half plane. [ABSTRACT FROM AUTHOR]

We make a complete wavelet analysis of asymptotic properties of distributions. The study is carried out via Abelian and Tauberian type results, connecting the boundary asymptotic behavior of the wavelet transform with local and non-local quasiasymptotic properties of elements in the Schwartz class of tempered distributions. Our Tauberian theorems are full characterizations of such asymptotic properties. We also provide precise wavelet characterizations of the asymptotic behavior of elements in the dual of the space of highly time-frequency localized functions over the real line. For the use of the wavelet transform in local analysis, we study the problem of extensions of distributions initially defined on ℝ∖{0} to ℝ; in this extension problem, we explore the asymptotic properties of extensions of a distribution having a prescribed asymptotic behavior. Our results imply intrinsic properties of functions and measures as well, for example, we give a new proof of the classical Littlewood Tauberian theorem for power series. [ABSTRACT FROM AUTHOR]

Abstract: We develop a distribution wavelet expansion theory for the space of highly time-frequency localized test functions over the real line and its dual space , namely, the quotient of the space of tempered distributions modulo polynomials. We prove that the wavelet expansions of tempered distributions converge in . A characterization of boundedness and convergence in is obtained in terms of wavelet coefficients. Our results are then applied to study local and non-local asymptotic properties of Schwartz distributions via wavelet expansions. We provide Abelian and Tauberian type results relating the asymptotic behavior of tempered distributions with the asymptotics of wavelet coefficients. [Copyright &y& Elsevier]

We establish some assertions of Tauberian and Abelian types which enable us to find connections between the asymptotic properties of the Laplace transform at infinity and the asymptotics of the corresponding densities of rapidly decaying distributions (at infinity or in some neighborhood of zero). As applications of our Tauberian type theorems we present asymptotics for the density f ( α,ρ)( x) of “extreme” stable laws with parameters ( α, ρ) for ρ = ±1 and x lying in the domain of rapid decay of f ( α,ρ)( x). This asymptotics had been found in [1–5] by a more complicated method. [ABSTRACT FROM AUTHOR]

This paper is concerned with recent developments on the Stieltjes transform of generalized functions. Sections 1 and 2 give a very brief introduction to the subject and the Stieltjes transform of ordinary functions with an emphasis to the inversion theorems. The Stieltjes transform of generalized functions is described in section 3 with a special attention to the inversion theorems of this transform. Sections 4 and 5 deal with the adjoint and kernel methods used for the development of the Stieltjes transform of generalized functions. The real and complex inversion theorems are discussed in sections 6 and 7. The Poisson transform of generalized functions, the iteration of the Laplace transform and the iterated Stieltjes transfrom are included in sections 8, 9 and 10. The Stieltjes transforms of different orders and the fractional order integration and further generalizations of the Stieltjes transform are discussed in sections 11 and 12. Sections 13, 14 and 15 are devoted to Abelian theorems, initial-value and final-value results. Some applications of the Stieltjes transforms are discussed in section 16. The final section deals with some open questions and unsolved problems. Many important and recent references are listed at the end.

Initial and final value Abelian theorems for the Whittaker transform of functions and of distributions are obtained. The Abelian theorems are obtained as the complex variable of the transform approaches 0 or ∞ in absolute value inside a wedge region in the right half plane.