Fourier series

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Diriclet Conditions

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

Theorem:

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

Orthogonal Functions

Orthonormal Functions
Weighing function
Kronecker delta function

The Fourier Series

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 relationships of the sine and cosine functions.