Spectral methods for polynomial interpolation and approximation

Classical problem of interpolation and approximation of functions with polynomials is considered here as a special case of spectral representation of functions. We developed this approach earlier for Legendre and Chebyshev orthogonal polynomials. Here we use Newton's fundamental polynomials as basis functions. We demonstrate that the spectral approach has computational advantages over the divided differences method. In a number of problems, Newton and Hermite interpolations are indistinguishable in our approach and computed by the same formulas. Also, computational algorithms that we constructed earlier with the use of orthogonal polynomials are adapted without modifications for use of Newton and Hermite polynomials.