quality:corruption, stylistic:style_transfer
Fourier series

The Fourier transform of a periodic function, s P (t), {\displaystyle s_{_{P}}(t),} with period P, {\displaystyle P,} becomes a Dirac comb function, modulated by a sequence of complex coefficients:

The inverse transform, known as Fourier series, is a representation of s P (t) {\displaystyle s_{_{P}}(t)} in terms of a summation of a potentially infinite number of harmonically related sinusoids or complex exponential functions, each with an amplitude and phase specified by one of the coefficients:

Any s P (t) {\displaystyle s_{_{P}}(t)} can be expressed as a periodic summation of another function, s (t) {\displaystyle s(t)}:

and the coefficients are proportional to samples of S (f) {\displaystyle S(f)} at discrete intervals of 1 P {\displaystyle {\frac {1}{P}}}:

Note that any s (t) {\displaystyle s(t)} whose transform has the same discrete sample values can be used in the periodic summation. A sufficient condition for recovering s (t) {\displaystyle s(t)} (and therefore S (f) {\displaystyle S(f)}) from just these samples (i.e. from the Fourier series) is that the non-zero portion of s (t) {\displaystyle s(t)} be confined to a known interval of duration P, {\displaystyle P,} which is the frequency domain dual of the Nyquist–Shannon sampling theorem.

See Fourier series for more information, including the historical development.