Book contents
- Frontmatter
- Contents
- Preface
- 1 p-adic numbers
- 2 p-adic functions
- 3 p-adic integration theory
- 4 p-adic distributions
- 5 Some results from p-adic ℒ1- and ℒ2-theories
- 6 The theory of associated and quasi associated homogeneous p-adic distributions
- 7 p-adic Lizorkin spaces of test functions and distributions
- 8 The theory of p-adic wavelets
- 9 Pseudo-differential operators on the p-adic Lizorkin spaces
- 10 Pseudo-differential equations
- 11 A p-adic Schrödinger-type operator with point interactions
- 12 Distributional asymptotics and p-adic Tauberian theorems
- 13 Asymptotics of the p-adic singular Fourier integrals
- 14 Nonlinear theories of p-adic generalized functions
- A The theory of associated and quasi associated homogeneous real distributions
- B Two identities
- C Proof of a theorem on weak asymptotic expansions
- D One “natural” way to introduce a measure on ℚp
- References
- Index
12 - Distributional asymptotics and p-adic Tauberian theorems
Published online by Cambridge University Press: 07 September 2011
- Frontmatter
- Contents
- Preface
- 1 p-adic numbers
- 2 p-adic functions
- 3 p-adic integration theory
- 4 p-adic distributions
- 5 Some results from p-adic ℒ1- and ℒ2-theories
- 6 The theory of associated and quasi associated homogeneous p-adic distributions
- 7 p-adic Lizorkin spaces of test functions and distributions
- 8 The theory of p-adic wavelets
- 9 Pseudo-differential operators on the p-adic Lizorkin spaces
- 10 Pseudo-differential equations
- 11 A p-adic Schrödinger-type operator with point interactions
- 12 Distributional asymptotics and p-adic Tauberian theorems
- 13 Asymptotics of the p-adic singular Fourier integrals
- 14 Nonlinear theories of p-adic generalized functions
- A The theory of associated and quasi associated homogeneous real distributions
- B Two identities
- C Proof of a theorem on weak asymptotic expansions
- D One “natural” way to introduce a measure on ℚp
- References
- Index
Summary
Introduction
Tauberian theorems is a generic name used to indicate results connecting the asymptotic behavior of a function (distribution) at zero with the asymptotic behavior of its Fourier, Laplace or other integral transforms at infinity; the inverse theorems are usually called abelian. In the real setting Tauberian theorems have numerous applications, in particular, in mathematical physics (for example, see Drozzinov and Zavyalov [80], [81], Korevaar [160], Nikolić-Despotović, Pilipović [191], Vladimirov, Drozzinov and Zavyalov [240], Yakymiv [248] and the references cited therein). Multidimensional Tauberian theorems for distributions are treated in the fundamental book [240]. Some of them are connected with the fractional operator. In [240], as a rule, theorems of this type are proved for distributions whose supports belong to a cone in ℝn (semi-axis for n = 1). This is related to the fact that such distributions form a convolution algebra. In this case the kernel of the fractional operator is a distribution whose support belongs to the cone in ℝn or a semi-axis for n = 1 [240, §2.8.]
p-adic analogs of Tauberian theorems do not seem to have been discussed so far except for [140], [141], [21]. In this chapter, we present a first study of them based on the above papers.
In the beginning, in Sections 12.2 and 12.3, we introduce the notion of the p-adic distributional asymptotics [140], [141]. In Section 12.2, the definition of distributional (stabilized) asymptotic estimate at infinity is introduced.
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- Theory of p-adic DistributionsLinear and Nonlinear Models, pp. 230 - 246Publisher: Cambridge University PressPrint publication year: 2010