A Distributional Theory for Asymptotic Expansions

Estrada, Ricardo; Kanwal, Ram P.

doi:10.1007/978-0-8176-8130-2_3

Cited by 31 publications

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“…1) The space K (R n ) plays an important role in the asymptotic analysis of distributions [6]. 2) Actually [4] given any sequence {µ k } k∈N n there exists a function λ ∈ S (R n ) such that a test function such that supp ϕ ⊂ (a, b) then f (y, x) , ϕ (x) x has compact support and vanishing moments, and thus f (y, x) , ϕ (x) x = 0.…”

Section: Several Facts About Distributionsmentioning

confidence: 99%

A two cones support theorem

Estrada¹

2016

OpMath

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Section: Several Facts About Distributionsmentioning

confidence: 99%

A two cones support theorem

Estrada¹

2016

OpMath

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“…The Schwartz spaces of test functions and distributions over the real line are denoted by D and D , respectively; the spaces of smooth rapidly decreasing functions and its dual, the space of tempered distributions, are denoted by S and S ; the spaces E and E are the test space of all C ∞ -functions and its dual, the space of compactly supported distributions; the spaces D + and S + denote the subspaces of D and S , respectively, consisting of distributions and tempered distributions with support in [0, ∞). We refer the reader to [9,17,24] for properties of these spaces.…”

Section: The Structure Of Quasiasymptoticsmentioning

confidence: 99%

“…The main subject of this section are the so-called quasiasymptotic behaviors of distributions at infinity and the origin which we now proceed to define [9,11,12,16,24,25,29]. There are several definitions in the literature, let us start with the one from [11,12].…”

Section: The Structure Of Quasiasymptoticsmentioning

confidence: 99%

“…In [13,11,12,14], the definition is extended to the form just presented here. It follows from the definition that ρ and g in (2.1) cannot have an arbitrary form [9,16,24]; indeed, ρ must be a regularly varying function [2,18] and g must be a homogeneous distribution [9] having degree of homogeneity equal to the index of regular variation of ρ.…”

Section: The Structure Of Quasiasymptoticsmentioning

confidence: 99%

“…for some constants γ and β, where here we are following the notation from [9]. When k = 1, we sometimes denote the distribution x −1 by p.v.…”

Section: The Structure Of Quasiasymptoticsmentioning

confidence: 99%

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The structure of quasiasymptotics of Schwartz distributions

Vindas

2010

Linear and Non-Linear Theory of Generalized Functions and Its Applications

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Abstract. In this article complete characterizations of quasiasymptotic behaviors 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 the quasiasymptotic of degree -1 and 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, the author presents a condition over test functions which allows one to evaluate them at the quasiasymptotic, these test functions are in bigger spaces than S. 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.

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Distributional radius of curvature

Estrada

2005

Math Methods in App Sciences

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SUMMARYWe show that any continuous plane path that turns to the left has a well-deÿned distribution that corresponds to the radius of curvature of smooth paths. We show that the distributional radius of curvature determines the path uniquely except for a translation. We show that Dirac delta contributions in the radius of curvature correspond to facets, that is, at sections of the path, and show how a path can be deformed into a facet by letting the radius of curvature approach a delta function. Copyright DIVISION BY ZEROEven in the very ÿrst stages of the teaching of mathematics, starting in elementary school, students are taught that one cannot divide by zero. This is important, of course, since division by zero is not well deÿned, and can yield non-sensical results.There are, however, several mathematical frameworks where one can consider the division by zero in a mathematically sound fashion. For instance, when studying indeterminate forms of limits in calculus, one uses the symbolic expression K=0 = ± ∞, which is a shorthand to the fact that if lim x→a f(x) = K = 0, and lim x→a g(x) = 0, with g(x) = 0 for x close to a but di erent from a, then lim x→a f(x)=g(x) = ± ∞, the sign being determined by the sign of K and the sign of g(x) near x = a. Similarly, in the theory of analytic functions it is convenient to consider the range of a meromorphic function to be C = C ∪ {∞}, the Riemann sphere, and deÿne the value of a meromorphic function at a pole to be ∞; thus, if f and g are analytic in a region , a ∈ , and f(a) = K = 0, while g(a) = 0, then the value of h(z) = f(z)=g(z) at z = a is ∞, which in a colourful way can be written as K=0 = ∞.Our aim in the present article is, however, to point out that in several cases the division by 0 could be interpreted not as ∞ but as a suitable Dirac delta function. That this is the * Correspondence to:

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A Distributional Theory for Asymptotic Expansions

Cited by 31 publications

References 0 publications

A two cones support theorem

A two cones support theorem

The structure of quasiasymptotics of Schwartz distributions

Distributional radius of curvature

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