Fourier series

Diriclet Conditions

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The conditions for a periodic function $f$ with period 2L to have a convergent Fourier series.

Theorem:

Let $f$ be a piecewise regular real-valued function defined on some interval [-L,L], such that $f$ has only a finite number of discontinuities and extrema in [-L,L]. Then the Fourier series of this function converges to $f$ when $f$ is continuous and to the arithmetic mean of the left-handed and right-handed limit of $f$ at a point where it is discontinuous.

The Fourier Series

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A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality (see Orthogonal Functions) relationships of the sine and cosine functions.

Principle author of this page: Aric Goe