### 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 [309, p. 325].
A direct proof can be based on the equivalance between Lagrange
interpolation and windowed-sinc interpolation using a ``scaled
binomial window'' [262,502]. That is,
for a fractional sample delay of samples, multiply the
shifted-by-, sampled, sinc function
sinc

by a binomial window
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