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 [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
by a binomial window
and normalize by 
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 .