图书简介

I Distributions and their Basic Applications./ 1 Basic Definitions and Operations./ 1.1 The \"delta function\" as viewed by a physicist and an engineer./ 1.2 A rigorous definition of distributions./ 1.3 Singular distributions as limits of regular functions./ 1.4 Derivatives; linear operations./ 1.5 Multiplication by a smooth function; Leibniz formula./ 1.6 Integrals of distributions; the Heaviside function./ 1.7 Distributions of composite arguments./ 1.8 Convolution./ 1.9 The Dirac delta on Rn, lines and surfaces./ 1.10 Linear topological space of distributions./ 1.11 Exercises./ 2 Basic Applications: Rigorous and Pragmatic./ 2.1 Two generic physical examples./ 2.2 Systems governed by ordinary differential equations./ 2.3 One/dimensional waves./ 2.4 Continuity equation./ 2.5 Green?s function of the continuity equation and Lagrangian coordinates./ 2.6 Method of characteristics./ 2.7 Density and concentration of the passive tracer./ 2.8 Incompressible medium./ 2.9 Pragmatic applications: beyond the rigorous theory of distributions./ 2.10 Exercises./ II Integral Transforms and Divergent Series./ 3 Fourier Transform./ 3.1 Definition and elementary properties./ 3.2 Smoothness, inverse transform and convolution./ 3.3 Generalized Fourier transform./ 3.4 Transport equation./ 3.5 Exercises./ 4 Asymptotics of Fourier Transforms./ 4.1 Asymptotic notation, or how to get a camel to pass through a needle?s eye./ 4.2 Riemann/Lebesgue Lemma./ 4.3 Functions with jumps./ 4.4 Gamma function and Fourier transforms of power functions./ 4.5 Generalized Fourier transforms of power functions./ 4.6 Discontinuities of the second kind./ 4.7 Exercises./ 5 Stationary Phase and Related Method./ 5.1 Finding asymptotics: a general scheme./ 5.2 Stationary phase method./ 5.3 Fresnel approximation./ 5.4 Accuracy of the stationary phase method./ 5.5 Method of steepest descent./ 5.6 Exercises./ 6 Singular Integrals and Fractal Calculus./ 6.1 Principal value distribution./ 6.2 Principal value of Cauchy integral./ 6.3 A study of monochromatic wave./ 6.4 The Cauchy formula./ 6.5 The Hilbert transform./ 6.6 Analytic signals./ 6.7 Fourier transform of Heaviside function./ 6.8 Fractal integration./ 6.9 Fractal differentiation./ 6.10 Fractal relaxation./ 6.11 Exercises./ 7 Uncertainty Principle and Wavelet Transforms./ 7.1 Functional Hilbert spaces./ 7.2 Time/frequency localization and the uncertainty principle./ 7.3 Windowed Fourier transform./ 7.4 Continuous wavelet transforms./ 7.5 Haar wavelets and multiresolution analysis./ 7.6 Continuous Daubechies? wavelets./ 7.7 Wavelets and distributions./ 7.8 Exercises./ 8 Summation of Divergent Series and Integrals./ 8.1 Zeno?s \"paradox\" and convergence of infinite series./ 8.2 Summation of divergent series./ 8.3 Tiring Achilles and the principle of infinitesimal relaxation./ 8.4 Achilles chasing the tortoise in presence of head winds./ 8.5 Separation of scales condition./ 8.6 Series of complex exponentials./ 8.7 Periodic Dirac deltas./ 8.8 Poisson summation formula./ 8.9 Summation of divergent geometric series./ 8.10 Shannon?s sampling theorem./ 8.11 Divergent integrals./ 8.12 Exercises./ A Answers and Solutions./ A.1 Chapter 1. Definitions and operations./ A.2 Chapter 2. Basic applications./ A.3 Chapter 3. Fourier transform./ A.4 Chapter 4. Asymptotics of Fourier transforms./ A.5 Chapter 5. Stationary phase and related methods./ A.6 Chapter 6. Singular integrals and fractal calculus./ A.7 Chapter 7. Uncertainty principle and wavelet transform./ A. 8 Chapter 8. Summation of divergent series and integrals./ B Bibliographical Notes.

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