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An Introduction to Fourier Series and Integrals

Robert T. Seeley
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a concise and mathematically rigorous introduction to Fourier analysis using Riemann integrals and some physical motivation. The exposition is driven by the Dirichlet problem: determining the steady-state heat distribution in a disk (Fourier series) or a half-plane (Fourier integrals) given the temperature on the boundary.

I like this approach because it leads to the positive Poisson kernel and to uniform rather than pointwise convergence, and gives an easy motivation for summability. The more common approach through the vibrating string and the non-positive Dirichlet kernel is a lot harder to do carefully and leaves the origin of summability a mystery.

The book concentrates on the heat conduction problem but also covers other traditional topics such as the vibrating string, convolution, Weierstrass's theorem on uniform approximation by polynomials, Bessel inequality, Parseval equality, and Plancherel formula. It has a lot of exercises, mostly to prove things, but a moderate number investigate particular Fourier series.

G. H. Hardy and W. W. Rogosinski's Cambridge Tract Fourier Series is another concise text but has very little in common with Seeley's book. They take a pure mathematical approach that uses the Lebesgue integral from the beginning. It is a traditional "real analysis" approach to Fourier series that focuses on Hilbert spaces and convergence and summability questions.

Dunham Jackson's Carus Monograph Fourier Series and Orthogonal Polynomials is somewhat in between these two books. It starts out as a pure mathematical approach, with the orthogonal sequences of trigonometric functions, and quickly moves to other eigenfunction expansions using Bessel functions and Legendre polynomials and many forms of orthogonal polynomials. It treats the physical problems as applications after the mathematical apparatus has been developed. It is really more about orthogonal polynomials than about Fourier analysis.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.




Editors' Foreword, iii

Preface, v


Introduction, 1


1. Dirichlet's Problem And Poisson's Theorem, 6

1-1. The Equation Of Steady State Heat Conduction, 7

1-2. The Solution Of The Differential Equation By Products, 10

1-3. The Fourier Coefficients, 13

1-4. Poisson's Kernel, 14

1-5. Assumption Of Boundary Values: Poisson's Theorem, 16

1-6. Two Simple Consequences Of Poisson's Theorem, 20

1-7. Uniqueness Of The Solution Of The Heat Problem, 21

1-8. The Pointwise Convergence Of Fourier Series, 24


2. The Method Of Separation Of Variables, 29

2-1. Sine And Cosine Series, 29

2-2. The Vibrating String, 33

2-3. Generalities On The Method Of Separation, 38


3. Some Applications Of Poisson's Theorem, 42

3-1. Uniform Approximation, 42

3-2. Least Squares Approximation, 45

3-3. Inner Products And Schwarz's Inequality, 48


4. Fourier Transforms, 57

4-1. Improper Integrals, 58

4-2. The Dirichlet Problem In A Half Plane, 65

4-3. Poisson's Kernel For A Half Plane, 68

4-4. The Maximum Principle For Harmonic Functions, And The Question Of Uniqueness, 72

4-5. Fourier Inversion And The Plancherel Formula, 77

4-6. Fourier Transforms And Derivatives, 84

4-7. Convolution, 87

4-8. The Time-Dependent Heat Equation, 91


Bibliography, 97

Answers To Exercises, 99

Index, 103