Chapter 12

Series and Transforms12.1 Fourier Series

A Fourier series is an expression of the form ∑∞k=−∞

ckeikx where this infinite sum is un-derstood to mean limn→∞ ∑

nk=−n ckeikx. Obviously such a sequence of partial sums may or

may not converge at a particular value of x.These series have been important in applied math since the time of Fourier who was an

officer in Napoleon’s army. He was interested in studying the flow of heat in cannons andinvented the concept to aid him in his study. Since that time, Fourier series and the mathe-matical problems related to their convergence have motivated the development of modernmethods in analysis.1 From the very beginning, the fundamental question has been relatedto the nature of convergence of these series. Dirichlet was the first to prove significant the-orems on this in 1829, but questions lingered till the mid 1960’s when a problem involvingconvergence of Fourier series was solved for the first time and the solution of this problemwas a big surprise.2 This chapter is on the classical theory of convergence of Fourier seriesstudied by Dirichlet, Riemann, and Fejer.

If you can approximate a function f with an expression of the form ∑∞k=−∞

ckeikx thenthe function must have the property f (x+2π) = f (x) because this is true of every term inthe above series. More generally, here is a definition.

Definition 12.1.1 A function f defined on R is a periodic function of period T iff (x+T ) = f (x) for all x.

As just explained, Fourier series are useful for representing periodic functions and noother kind of function.There is no loss of generality in studying only functions which areperiodic of period 2π . Indeed, if f is a function which has period T , you can study thisfunction in terms of the function g(x)≡ f

(T x2π

)where g is periodic of period 2π .

Definition 12.1.2 For f ∈ R([−π,π]) and f periodic on R, define the Fourier se-ries of f as

∞

∑k=−∞

ckeikx, (12.1)

whereck ≡

12π

∫π

−π

f (y)e−ikydy. (12.2)

Also define the nth partial sum of the Fourier series of f by

Sn ( f )(x)≡n

∑k=−n

ckeikx. (12.3)

1Fourier was with Napoleon in Egypt when the Rosetta Stone was discovered and wrote about Egypt in De-scription de l’Égypte. He was a teacher of Champollion who eventually made it possible to read Egyptian byusing the Rosetta Stone, discovered at this time. This expedition of Napoleon caused great interest in all thingsEgyptian in the first part of the nineteenth century.

2The question was whether the Fourier series of a function in L2 converged almost everywhere to the functionwhere the term “almost everywhere” has a precise meaning. (L2 means the square of the function is integrable.)It turned out that it did, to the surprise of many because it was known that the Fourier series of a function in L1

does not necessarily converge to the function almost everywhere, this from an example given by Kolmogorov inthe 1920’s. The problem was solved by Carleson in 1965.

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