A Distributional Approach to Asymptotics Theory and Applications

"...The authors of this remarkable book are among the very few who have faced up to the challenge of explaining what an asymptotic expansion is, and of systematizing the handling of asymptotic series. The idea of using distributions is an original one, and we recommend that you read the book......

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Bibliographic Details
Main Authors: Estrada, Ricardo, Kanwal, Ram P. (Author)
Format: eBook
Language:English
Published: Boston, MA Birkhäuser 2002, 2002
Edition:2nd ed. 2002
Series:Birkhäuser Advanced Texts Basler Lehrbücher
Subjects:
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 7.4 Dual Taylor Series as Asymptotics of Solutions of Equations
  • 7.5 Boundary Layers
  • 7.6 Spectral Content Asymptotics
  • 7.7 Exercises
  • References
  • 3.2 The Taylor Expansion of Distributions
  • 3.3 The Moment Asymptotic Expansion
  • 3.4 Expansions in the Space P?
  • 3.5 Laplace’s Asymptotic Formula
  • 3.6 The Method of Steepest Descent
  • 3.7 Expansion of Oscillatory Kernels
  • 3.8 Time-Domain Asymptotics
  • 3.9 The Expansion of f (?x) as ? ? ? in Other Cases
  • 3.10 Asymptotic Separation of Variables
  • 3.11 Exercises
  • 4 Asymptotic Expansion of Multidimensional Generalized Functions
  • 4.1 Introduction
  • 4.2 Taylor Expansion in Several Variables
  • 4.3 The Multidimensional Moment Asymptotic Expansion
  • 4.4 Laplace’s Asymptotic Formula
  • 4.5 Fourier Type Integrals
  • 4.6 Time-Domain Asymptotics
  • 4.7 Further Examples
  • 4.8 Tensor Products and Partial Asymptotic Expansions
  • 4.9 An Application in Quantum Mechanics
  • 4.10 Expansion of Kernels of the Type f (?x, x)
  • 4.11 Exercises
  • 5 Asymptotic Expansion of Certain Series Considered by Ramanujan
  • 5.1 Introduction
  • 5.2 Basic Formulas
  • 5.3 Lambert Type Series
  • 5.4 Distributionally Small Sequences
  • 5.5 Multiple Series
  • 5.6 Unrestricted Partitions
  • 5.7 Exercises
  • 6 Cesàro Behavior of Distributions
  • 6.1 Introduction
  • 6.2 Summability of Series and Integrals
  • 6.3 The Behavior of Distributions in the (C) Sense
  • 6.4 The Cesàro Summability of Evaluations
  • 6.5 Parametric Behavior
  • 6.6 Characterization of Tempered Distributions
  • 6.7 The Space K?
  • 6.8 Spherical Means
  • 6.9 Existence of Regularizations
  • 6.10 The Integral Test
  • 6.11 Moment Functions
  • 6.12 The Analytic Continuation of Zeta Functions
  • 6.13 Fourier Series
  • 6.14 Summability of Trigonometric Series
  • 6.15 Distributional Point Values of Fourier Series
  • 6.16 Spectral Asymptotics
  • 6.17 Pointwise and Average Expansions
  • 6.18 Global Expansions
  • 6.19 Asymptotics of the Coincidence Limit
  • 6.20 Exercises
  • 7 Series of Dirac Delta Functions
  • 7.1 Introduction
  • 7.2 Basic Notions
  • 7.3 Several Problems that Lead to Series of Deltas
  • 1 Basic Results in Asymptotics
  • 1.1 Introduction
  • 1.2 Order Symbols
  • 1.3 Asymptotic Series
  • 1.4 Algebraic and Analytic Operations
  • 1.5 Existence of Functions with a Given Asymptotic Expansion
  • 1.6 Asymptotic Power Series in a Complex Variable
  • 1.7 Asymptotic Approximation of Partial Sums
  • 1.8 The Euler-Maclaurin Summation Formula
  • 1.9 Exercises
  • 2 Introduction to the Theory of Distributions
  • 2.1 Introduction
  • 2.2 The Space of Distributions D?
  • 2.3 Algebraic and Analytic Operations
  • 2.4 Regularization, Pseudofunction and Hadamard Finite Part
  • 2.5 Support and Order
  • 2.6 Homogeneous Distributions
  • 2.7 Distributional Derivatives of Discontinuous Functions
  • 2.8 Tempered Distributions and the Fourier Transform
  • 2.9 Distributions of Rapid Decay
  • 2.10 Spaces of Distributions Associated with an Asymptotic Sequence
  • 2.11 Exercises
  • 3 A Distributional Theory for Asymptotic Expansions
  • 3.1 Introduction