Fourier transformation

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Fourier transformation is an invertible integral transformation that decompose a square integrable, piecewise continuous function on a topological group into a linear combination of basis elements.

Often, functions which are difficult to analyze in one topological group become much easier to analyze when transformed to another topological group.

Discrete Fourier transformation

Discrete Fourier transformations are defined on discrete topological groups, and the integral is replaced by summation.