Polynomial approximation with Pollaczek–type weights. A survey

Abstract

The paper deals with weighted polynomial approximation for functions defined on (−1,1), which can grow exponentially both at −1 and at 1. We summarize recent results on function spaces with new moduli of smoothness, estimates for the best approximation, Lagrange interpolation, Fourier sums and Gaussian rules with respect to weights of the form w(x)=(1−x2)βe−(1−x2)−α.