Spectral methods of polynomial interpolation and approximation

The classical problem of interpolation and approximation of functions by polynomials is considered here as a special case of spectral representation of functions. We have previously developed this approach for the Legendre and Chebyshev orthogonal polynomials. Here we use fundamental Newton polynomials as basis functions. It is shown that the spectral approach has computational advantages over the divided difference method. In a number of problems, Newton and Hermite interpolations are indistinguishable with our approach and are calculated by the same formulas. Also, the computational algorithms we proposed earlier using orthogonal polynomials are transferred without changes to Newton and Hermite polynomials.