Relation of Lagrange and Sinc Interpolation

For an infinite number of equally spaced samples, with spacing , the Lagrangian basis polynomials converge to shifts of the sinc function, i.e.,

where

A simple argument is based on the fact that any analytic function is determined by its zeros and its value at one point. Since sinc is zero on all the integers except 0, and since sinc, it must coincide with the infinite-order Lagrangian basis polynomial for the sample at which also has its zeros on the nonzero integers and equals at .

The equivalence of sinc interpolation to Lagrange interpolation was apparently first published by the mathematician Borel in 1899, and has been rediscovered many times since [318, p. 325].

A direct proof can be based on the equivalance between Lagrange interpolation and windowed-sinc interpolation using a ``binomial window'' [269,515]. That is, for a fractional sample delay of samples, multiply the shifted-by-, sampled, sinc function

by a binomial window

and normalize by [515]

which normalizes the interpolating filter to have a unit norm, to obtain the th-order Lagrange interpolating filter

Since the binomial window converges to the Gaussian window as , and since the window gets wider and wider, approaching a unit constant in the limit, the convergence of Lagrange to sinc interpolation can be seen.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.