Polynomial approximation with an exponential weight in [−1, 1] (revisiting some of Lubinsky’s results)

Abstract

Revisiting the results in [7], [8], we consider the polynomial approximation on (−1, 1) with the weight \(w(x) = e^{-(1-x^2)^{-\alpha}}\), α > 0. We introduce new moduli of smoothness, equivalent to suitable K-functionals, and we prove the Jackson theorem, also in its weaker form. Moreover, we state a new Bernstein inequality, which allows us to prove the Salem-Stechkin inequality. Finally, also the behaviour of the derivatives of the polynomials of best approximation is discussed.