In mathematics the finite Fourier transform may refer to either

another name for the discrete Fourier transform^[1]

another name for the Fourier series coefficients^[2]

a transform based on a Fourier-transform-like integral applied to a function <math>x(t)</math>, but with integration only on a finite interval, usually taken to be the interval <math>[0,T]</math>.^[3] Equivalently, it is the Fourier transform of a function <math>x(t)</math> multiplied by a rectangular window function. That is, the finite Fourier transform <math>X(\omega)</math> of a function <math>x(t)</math> on the finite interval <math>[0,T]</math> is given by: