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Fourier series
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Fourier series

 

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 ei 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 Parseval's theorem
5 General formulation
6 See also
7 References

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

In the important special case of a real-valued function f(x), one often uses the identity

to equivalently represent f(x) as a infinite linear combination of functions of the form cos(nx) and sin(nx), i.e.

which corresponds to and .

Convergence of Fourier series

While the coefficients an and bn 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.

The simplest answer is that if f is square-integrable then

(this is convergence in the norm of the space L2).

There are also many known tests that ensure that the series converges at a given point x. For example, if the function is differentiable at x. Even a jump discontinuity does not pose a problem: if the function has left and right derivatives at x, then the Fourier series will converge to the average of the left and right limits (but see Gibbs phenomenon). It is also known that for any function of any Hölder class and any function of bounded variation the Fourier series converges everywhere.

However, a fact that many find surprising, is that the Fourier series of a continuous function need not converge pointwise. The easiest proof is already nontrivial since it appeals to the Banach-Steinhaus uniform boundedness principle and is nonconstructive (that is, it shows that a continuous function whose Fourier series does not converge at 0 does exist without actually saying what that function might look like). The same argument also allow to give a constructive proof, though the function is presented as an infinite sum, and not in a simple formula. There are various related proofs showing stronger results (for instance, that there is an ample supply of such continuous functions, or that there are continuous functions such that the Fourier series fails to converge pointwise on any given set of measure zero).

The problem whether the Fourier series of any continuous function converges almost everywhere was posed by Nikolai Lusin in the 1920s and remained open until finily resolved positively in 1966 by Lennart Carleson. Indeed, Carleson showed that the Fourier expansion of any function in L2 cnverges almost everywhere. Later on Hunt generalized this to Lp for any p>1. Despite a number of attempts at simplifying the proof, it is still one of the most difficult results in analysis. Contrariwise, Kolmogorov, in his very first paper published when he was 21, constructed an example of a function in L1 whose Fourier series diverges almost everywhere (later improved to divergence everywhere).

Additional research on these problems contains

Some positive consequences of the homomorphism properties of exp

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

then (if G is the transform of g)

that is, the Fourier transform of a convolution is the product of the Fourier transforms. Vice versa, if then the Fourier transform H of h is the convolution of the Fourier transforms of f and g

Parseval's theorem

Another important property of the Fourier series is Parseval's theorem, a special case of the Plancherel theorem and a form of unitarity:

or, for the real-valued f(x) case above,

General formulation

The useful properties of Fourier series are largely derived from the orthogonality and homomorphism property of the functions ei 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.

See also

References

Textbooks

The Katznelson book is the one using the most modern terminology and style of the three. The original publishing dates are: Zygmund in 1935, Bari in 1961 and Katznelson in 1968.

Articles referred to in the text

This is the first proof that the Fourier series of a continuous function might diverge. In German The first is a construction of an integrable function whose Fourier series diverges almost everywhere. The second is a strengthening to divergance everywhere. In French. This is the original paper of Carleson, where he proves that the Fourier expansion of any continuous function converges almost everywhere, and two attempts at simplifying the proof. In this paper the authors show that for any set of zero measure there exists a continuous function on the circle whose Fourier series diverges on that set. In French. In this paper the author improves on Kolmogorov's result by constructing an integrable function with the Fourier series diverging fast everywhere. This result is the best known, but it is not known whether it is sharp.