The normalized sinc function is the Fourier transform of the rectangular function with no scaling. It is used in the concept of reconstructing a continuous bandlimited signal from uniformly spaced samples of that signal.

The sinc function: x ( t ) = sin ( a t ) t , a > 0 . As can be expected, its FCT behaves much like a step function, (2.20) X c ( ω ) = ( 2 π ) 1 / 2 ∫ 0 ∞ sin ( a t ) t cos ( ω t ) d t = ( π 2 ) 1 / 2 , if ω < a , = 1 2 ( π 2 ) 1 / 2 , if ω = a , = 0 , otherwise .

The cosc function is obviously an odd function but its magnitude Fourier transform is the same as that of the sinc function. A combination of these two function creates a basis function that spans the band-limited functions in (Paley–Wiener space).