Author | Estrada, Ricardo. author |
---|---|
Title | A Distributional Approach to Asymptotics [electronic resource] : Theory and Applications / by Ricardo Estrada, Ram P. Kanwal |
Imprint | Boston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2002 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-0-8176-8130-2 |
Descript | XV, 454 p. online resource |
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 -- 3.2 The Taylor Expansion of Distributions -- 3.3 The Moment Asymptotic Expansion -- 3.4 Expansions in the Space P? -- 3.5 Laplaceโs 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 Laplaceโs 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 Cesร ro 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 Cesร ro 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 -- 7.4 Dual Taylor Series as Asymptotics of Solutions of Equations -- 7.5 Boundary Layers -- 7.6 Spectral Content Asymptotics -- 7.7 Exercises -- References