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Lagrange Interpolation

• Lagrange interpolation is just polynomial interpolation
• th-order polynomial interpolates points
• First-order case = linear interpolation

Problem Formulation

Given a set of known samples , , find the unique order polynomial which interpolates the samples

Solution (Waring, Lagrange): where is the Lagrange polynomial corresponding to sample : • Numerator gives a zero at all samples but the th
• Denominator simply normalizes to at • As a result, • Generalized bandlimited impulse = generalized sinc function:
Each goes through at and zero at all other sample points
I.e., is analogous to sinc • For uniformly spaced samples, Lagrange interpolaton converges to sinc interpolation as • For uniformly spaced samples and finite , Lagrange interpolaton is equivalent to windowed sinc interpolation using a binomial window
(see text for refs)

• Nonuniformly spaced sample locations, such as along the zeros of a Chebyshev polynomial, generally do better than uniform spacing, when applicable

Subsections
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