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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 [291, p. 325].
A direct proof can be based on the equivalance between Lagrange
interpolation and windowed-sinc interpolation using a ``binomial
window'' [245,477]. That is, for a
fractional sample delay of
samples, multiply the shifted-by-
,
sampled, sinc function
![$\displaystyle h_s(n) =$](img2165.png)
sinc
by a binomial window
and normalize by [477]
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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