Asymptotic Behavior of Generalized Functions

The following sections are included:

• Preface

• Contents

The following sections are included:

• Preliminaries

• S-asymptotics in
  • Definition
  • Characterization of comparison functions and limits
  • Equivalent definitions of the S-asymptotics in
  • Basic properties of the S-asymptotics
  • S-asymptotic behavior of some special classes of generalized functions
    • Examples with regular distributions
    • Examples with distributions in subspaces of
    • S-asymptotics of ultradistributions and Fourier hyperfunctions — Comparisons with the S-asymptotics of distributions
  • S-asymptotics and the asymptotics of a function
  • Characterization of the support of
  • Characterization of some generalized function spaces
  • Structural theorems for S-asymptotics in
  • S-asymptotic expansions in
    • General definitions and assertions
    • S-asymptotic Taylor expansion
  • S-asymptotics in subspaces of distributions
  • Generalized S-asymptotics

• Quasi-asymptotics in
  • Definition of quasi-asymptotics at infinity over a cone
  • Basic properties of quasi-asymptotics over a cone
  • Quasi-asymptotic behavior at infinity of some generalized functions
  • Equivalent definitions of quasi-asymptotics at infinity
  • Quasi-asymptotics as an extension of the classical asymptotics
  • Relations between quasi-asymptotics in and
  • Quasi-asymptotics at ± ∞
  • Quasi-asymptotics at the origin
  • Quasi-asymptotic expansions
  • The structure of quasi-asymptotics. Up-to-date results in one dimension
    • Remarks on slowly varying functions
    • Asymptotically homogeneous functions
    • Relation between asymptotically homogeneous functions and quasi-asymptotics
    • Associate asymptotically homogeneous functions
    • Structural theorems for negative integral degrees. The general case
  • Quasi-asymptotic extension
    • Quasi-asymptotics at the origin in and
    • Quasi-asymptotic extension problem in
    • Quasi-asymptotics at infinity and spaces
  • Quasi-asymptotic boundedness
  • Relation between the S-asymptotics and quasi-asymptotics at ∞

The following sections are included:

• Asymptotic behavior of solutions to partial differential equations
  • S-asymptotics of solutions
  • Quasi-asymptotics of solutions
  • S-asymptotics of solutions to equations with ultra-differential or local operators

• Asymptotics and integral transforms
  • Abelian type theorems
    • Transforms with general kernels
    • Special integral transforms
  • Tauberian type theorems
    • Convolution type transforms in spaces of distributions
    • Convolution type transforms in other spaces of generalized functions
    • Integral transforms of Mellin convolution type
    • Special integral transforms
    • Localization of Tauberian type theorems

• Summability of Fourier series and integrals
  • The Cesàro behavior
    • Cesàro, Riesz, and Abel summability of series and integral
    • The Cesàro behavior of distributions
    • Summability of distributional evaluations
  • Summability of the Fourier transform and distributional point values
    • Characterization of distributional point values of tempered distributions
    • Abel summability
    • Convergence of Fourier Series
    • Series with gaps
    • Convergence of Fourier integrals
    • Tauberian theorem for distributional point values
    • Application: Littlewood's Tauberian theorem
    • Solution to the Hardy-Littlewood (C) summability problem for distributions

The following sections are included:

• Bibliography

• Index