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Fourier Series

Rajendra Bhatia
Mathematical Association of America
Publication Date: 
Number of Pages: 
Classroom Resource Materials
[Reviewed by
Brandy J. Lipton
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Bhatia’s textbook, Fourier Series, makes two important claims in its preface — that it will convey a sense of the importance and applicability of Fourier analysis to the reader and that it is accessible, for the most part, to a third year undergraduate student of mathematics. Bhatia attempts to fulfill these assurances by including a section on the sine and cosine functions containing ample examples and graphs, numerous historical references, and a concluding chapter on the applications of Fourier analysis while drawing on the reader’s knowledge of real analysis and differential equations. Bhatia does make references to complex analysis, and chapter four of his text looks more like the second course in real analysis than the first (Bhatia does alert the reader to the more advanced material found in chapter four in his preface), but for the most part the text does seem appropriate for the stated audience.

Bhatia opens with a chapter discussing the historical discoveries related to Fourier analysis. He organizes the history by mathematician rather than chronologically, which seems fitting in the context of Fourier analysis since each contributing mathematician’s work is so strongly linked to the work his predecessors. The mathematics is heavily interspersed in Bhatia’s history chapter, as the history is later interspersed with the mathematical chapters of the text.

Bhatia then gives the reader a standard introduction to the Laplace equation in chapter one. Notably he describes the heat equation, but does not discuss waves. This chapter progresses quickly, introducing the reader to convolutions, Dirac sequences, and good kernels. Bhatia proves some of the basic theorems, but the information in this chapter seems extremely compact. This aspect of the text seems to suggest that it would be more suitable as a supplementary text than as the main textbook for an undergraduate level course. Bhatia’s chapter on pointwise convergence, on the other hand, is more elaborate. He begins integrating rich descriptions of the mathematics into the text, especially in his explanations of the principle of localization and the method of condensation of singularities. His proof of the Du-Bois Reymond Theorem using the Uniform Boundedness Principle in this chapter is also very well done.

In chapters three and five, Bhatia piques the reader’s interest with multiple examples, graphs, and interesting applications. His section on π and infinite series and his description of the Gibbs phenomenon are particularly nice. Finally, his chapter on L1 and L2 convergence is very thorough. He gives multiple examples of Hilbert spaces, and most importantly, Bhatia relates the proof that the Fourier series of a continuous function is always Cesaro summable to that function in the L1 norm.

Overall, Bhatia is able to derive the important results of introductory Fourier Analysis in a readable way. His presentation is interesting, and his many short detours into related issues inspire the reader to delve deeper into analysis. Although certain “basics” of the subject are described quickly and in relatively little detail, Bhatia’s text could certainly serve as a complementary undergraduate resource or as the main textbook in a preparatory graduate course.

 Brandy Lipton is a graduate student in Economics at Northwestern University; she wrote this review when she was a senior mathematics major at Colby College.

History of Fourier Series

Heat Conduction and Fourier Series:
The Laplace equation in two dimensions
Solutions of the Laplace equation
The complete solution of the Laplace equation

Convergence of Fourier Series:
Abel summability and Cesaro summability
The Dirichlet and Fejér kernels
Pointwise Convergence of Fourier series
Term by term integration and differentiation
Divergence of Fourier series

Odds and Ends:
Sine and cosine series
Functions with arbitrary periods
Some simple examples
Infinite products
\(\pi\) and infinite series
Bernoulli numbers
The Gibbs phenomenon
A historical digression

Convergence in \(L_2\) and \(L_1\): 
\(L_2\)-convergence of Fourier series
Fourier coefficients of \(L_1\) 

Some Applications:
ergodic theorem and number theory
The isoperimetric problem
The vibrating string
Band matrices