Background material

Fourier analysis

Let f be a function defined on the real numbers. We say that f is periodic with a period T if f fulfils the equation f(x+T)=f(x) for all real numbers x, where T is the smallest number with this property. Typical examples of periodic functions are sin(kx) and cos(kx).

Theorem

Let f be continuous on I=[-π, π]. Assume that the series

converges uniformly to the function f on the interval I. Then we have

Definition

The coefficients a_n and b_n are called Fourier coefficients of f, and the series

is called the Fourier series of f.

Example

Let f be the function that equals 1 in the interval [0, π] and –1 in [-π, 0]. Then we have

when n is odd and b_n=0 for n jevn. Det gir Fourier-rekken

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