### 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

A more recent alternate proof appears in [557].

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