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.,






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

![$\displaystyle (n-D) = \frac{\sin[\pi(n-D)]}{\pi(n-D)}
$](http://www.dsprelated.com/josimages_new/pasp/img1113.png)






A more recent alternate proof appears in [557].
Next Section:
Thiran Allpass Interpolation in Matlab
Previous Section:
Recent Developments in Lagrange Interpolation