傅里葉分析及其應(yīng)用

內(nèi)容概要

A carefully prepared account of the
basic ideas in Fourier analysis and its applications to the study
of partial differential equations. The author succeeds to make his
exposition accessible to readers with a limited background, for
example, those not acquainted with the Lebesgue integral. Readers
should be familiar with calculus, linear algebra, and complex
numbers. At the same time, the author has managed to include
discussions of more advanced topics such as the Gibbs phenomenon,
distributions, Sturm-Liouville theory, Cesaro summability and
multi-dimensional Fourier analysis, topics which one usually does
not find in books at this level. A variety of worked examples and
exercises will help the readers to apply their newly acquired
knowledge.

作者簡介

作者：（瑞典）弗雷特布拉德（Anders Vretblad）

書籍目錄

preface 
1 introduction 
1.1 the classical partial differential equations 
1.2 well-posed problems 
1.3 the one-dimensional wave equation 
1.4 fourier's method 
2 preparations 
2.1 complex exponentials 
2.2 complex-valued functions of a real variable 
2.3 cesaro summation of series 
2.4 positive summation kernels 
2.5 the riemann-lebesgue lemma 
2.6 *some simple distributions 
2.7 *computing with δ 
3 laplace and z transforms 
3.1 the laplace transform 
3.2 operations 
3.3 applications to differential equations 
3.4 convolution 
.3.5 *laplace transforms of distributions 
3.6 the z transform 
3.7 applications in control theory 
summary of chapter 3 
4 fourier series 
4.1 definitions 
4.2 dirichlet's and fejer's kernels; uniqueness 
4.3 differentiable functions 
4.4 pointwise convergence 
4.5 formulae for other periods 
4.6 some worked examples 
4.7 the gibbs phenomenon 
4.8 *fourier series for distributions 
summary of chapter 4 
5 l2 theory 
5.1 linear spaces over the complex numbers 
5.2 orthogonal projections 
5.3 some examples 
5.4 the fourier system is complete 
5.5 legendre polynomials 
5.6 other classical orthogonal polynomials 
summary of chapter 5 
6 separation of variables 
6.1 the solution of fourier's problem 
6.2 variations on fourier's theme 
6.3 the dirichlet problem in the unit disk 
6.4 sturm-liouville problems 
6.5 some singular sturm-liouville problems 
summary of chapter 6 
7 fourier transforms 
7.1 introduction 
7.2 definition of the fourier transform 
7.3 properties 
7.4 the inversion theorem. 
7.5 the convolution theorem 
7.6 plancherel's formula 
7.7 application i 
7.8 application 2 
7.9 application 3: the sampling theorem 
7.10 *connection with the laplace transform 
7.11 *distributions and fourier transforms 
summary of chapter 7 
8 distributions 
8.1 history 
8.2 fuzzy points - test functions 
8.3 distributions 
8.4 properties 
8.5 fourier transformation 
8.6 convolution 
8.7 periodic distributions and fourier series 
8.8 fundamental solutions 
8.9 back to the starting point 
summary of chapter 8 
9 multi-dimensional fourier analysis 
9.1 rearranging series 
9.2 double series 
9.3 multi-dimensional fourier series 
9.4 multi-dimensional fourier transforms 
appendices 
a the ubiquitous convolution 
b the discrete fourier transform 
c formulae 
c.1 laplace transforms 
c.2 z transforms 
c.3 fourier series 
c.4 fourier transforms 
c.5 orthogonal polynomials 
d answers to selected exercises 
e literature 
index

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