Fourier series

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In mathematics, a Fourier series, named in honor of Joseph Fourier (1768-1830), is a representation of a periodic function as a sum of periodic functions of the form

which are harmonics of e^{i x}; Fourier was the first to study such series systematically. He applied these series to the solution of the heat equation, publishing his initial results in 1807 and 1811. This area of inquiry is sometimes called harmonic analysis.

Many other Fourier-related transforms have since been defined.

Table of contents

1 Definition of Fourier series
2 Convergence of Fourier series
3 Some positive consequences of the homomorphism properties of exp
4 General formulation
5 See also

Definition of Fourier series

Suppose f(x) is a complex-valued function of a real number, is periodic with period 2π, and is square-integrable over the interval from 0 to 2π. Let

Then the Fourier series representation of f(x) is given by

Since

this is equivalent to representing f(x) as a infinite linear combination of functions of the form cos(nx) and sin(nx), i.e.

Convergence of Fourier series

While the coefficients a_n and b_n can be formally defined for any function for which the integrals make sense, whether the series so defined actually converges to f(x) depends on the properties of f.

A partial answer is that if f is square-integrable then

(this is convergence in the norm of the space).

That much was proved in the 19th century, as was the fact that if f is piecewise continuous then the series converges at each point of continuity. Perhaps surprisingly, it was not shown until the 1960s that if f is quadratically integrable then the series converges for every value of x except those in some set of measure zero.

Some positive consequences of the homomorphism properties of exp

Because "basis functions" e^ikx are homomorphisms of the real line (more precisely, of the "circle group") we have some useful identities:

If then
The Fourier transform is a morphism: -- that is, the Fourier transform of a convolution is the product of the Fourier transforms.

General formulation

The useful properties of Fourier series are largely derived from the orthogonality and homomorphism property of the functions e^{i n x}. Other sequences of orthogonal functions have similar properties, although some useful identities concerning e.g. convolutions are no longer true once we lose the homomorphism property. Examples include sequences of Bessel functions and orthogonal polynomials Such sequences are commonly the solutions of a differential equation; a large class of useful sequences are solutions of the so-called Sturm-Liouville problems.