Cosine transform

In mathematics, the Fourier sine and cosine transforms are forms of the Fourier integral transform that do not use complex numbers. They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or statistics.

The Fourier sine transform of  f (t), sometimes denoted by either ${\displaystyle {\hat {f}}^{s}}$ or ${\displaystyle {\mathcal {F}}_{s}(f)}$, is

If t means time, then ν is frequency in cycles per unit time, but in the abstract, they can be any pair of variables which are dual to each other.

This transform is necessarily an odd function of frequency, i.e. for all ν:

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