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.

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