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The structure of quasiasymptotics of Schwartz distributions

100%
Vindas J.
Banach Center Publications
|
2010
|
tom 88
|
nr 1
297-314
EN
In this article complete characterizations of the quasiasymptotic behavior of Schwartz distributions are presented by means of structural theorems. The cases at infinity and the origin are both analyzed. Special attention is paid to quasiasymptotics of degree -1. It is shown how the structural theorem can be used to study Cesàro and Abel summability of trigonometric series and integrals. Further properties of quasiasymptotics at infinity are discussed. A condition for test functions in bigger spaces than 𝓢 is presented which allows one to consider the respective quasiasymptotics over them. An extension of the structural theorems for quasiasymptotics is given. The author studies a structural characterization of the behavior f(λx) = O(ρ(λ)) in 𝒟', where ρ is a regularly varying function.
2
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The prime number theorem for Beurling's generalized numbers. New cases

63%
Schlage-Puchta J.-C., Vindas J.
Acta Arithmetica
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2012
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tom 153
|
nr 3
299-324
3
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Distributionally regulated functions

63%
Vindas J., Estrada R.
Studia Mathematica
|
2007
|
tom 181
|
nr 3
211-236
EN
We study the class of distributions in one variable that have distributional lateral limits at every point, but which have no Dirac delta functions or derivatives at any point, the "distributionally regulated functions." We also consider the related class where Dirac delta functions are allowed. We prove several results on the boundary behavior of functions of two variables F(x,y), x ∈ ℝ, y>0, with F(x,0⁺) = f(x) distributionally, both near points where the distributional point value exists and points where the lateral distributional limits exist. We give very general formulas for the jumps, in terms of F and related functions. We prove that the set of singular points of a distributionally regulated function is always countable at the most. We also characterize the Fourier transforms of tempered distributionally regulated functions in two ways.
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