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

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'' [264,506]. That is, for a fractional sample delay of samples, multiply the shifted-by- , sampled, sinc function

sinc

by a binomial window

and normalize by [506]

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 [561].

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