Lagrange interpolation is a well known, classical technique for
interpolation [7]. It is also called Waring-Lagrange
interpolation, since Waring actually published it 16 years before
Lagrange [11, p. 323]. Given a set of *n*+1 known
samples *f*(*x*_{k}),
, the problem is to find the
unique order *n* polynomial *y*(*x*) which interpolates the samples.
The solution can be expressed as a linear combination of elementary
*n*th order polynomials:

(17) |

(18) |

(19) |

In the case of an *infinite *number of *equally spaced *samples, with spacing
, the Lagrangian basis
polynomials converge to shifts of the *sinc function, *i.e.,

(20) |

A simple argument is based on the fact that any analytic function is determined by its zeros and its value at one point. Since is zero on all the integers except 0, and since , it must coincide with the infinite-order Lagrangian basis polynomial for the sample 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 [11, p. 325].

A direct proof can be based on the equivalance between Lagrange
interpolation and windowed-sinc interpolation using a ``scaled
binomial window'' [9,23]. That is,
for a fractional sample delay of *D* samples, multiply the
shifted-by-*D*, sampled, sinc function

by a binomial window

and normalize by [23]

which scales the interpolating filter to have a unit

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 [24].

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