Lp Norms and the Sinc Function

 sinx x if x 6= 0 1 otherwise This function is important in many areas of computing science, approximation theory, and numerical analysis. For example, it is used in interpolation and approximation of functions, approximate evaluation of transforms (e.g. Hilbert, Fourier, Laplace, Hankel, and Mellon transforms as well as the fast Fourier transform). It is used in finding approximate solutions of differential and integral equations, in image processing (it is the Fourier transform of the box filter and central to the understanding of the Gibbs phenomenon [12]), in signal processing and information theory. Much of this is nicely described in [7]. The first explicit appearance of the sinc function in approximation theory was probably in the use of the Whittaker cardinal functions C(f, h) to approximate functions analytic on an interval or on a contour. Given a function f which is defined on the real line R, the function C(f, h) is defined by C(f, h) = ∞ ∑