Relation of Lagrange Interpolation to Windowed Sinc Interpolation

• For an infinite number of equally spaced samples, with spacing $x_{k+1}-x_k = \Delta$ , the Lagrange-interpolation basis polynomials converge to shifts of the sinc function, i.e.,
  $$l_k(x) =$$   sinc$$\left(x-k\Delta\over\Delta\right), \quad k=\ldots,-2,-1,0,1,2,\ldots$$
  Proof: As order $\to\infty$ , the binomial window $\to$ Gaussian window $\to$ constant (unity).