Relation of Lagrange to 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

sinc

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 [312, p. 325].

A direct proof can be based on the equivalance between Lagrange interpolation and windowed-sinc interpolation using a ``scaled binomial window'' [264,504]. That is, for a fractional sample delay of samples, multiply the shifted-by- , sampled, sinc function

sinc

by a binomial window

and normalize by [504]

which scales 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.

A more recent alternate proof appears in [559].

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