Best one-sided approximation and optimal Hermite–Fejér interpolation on an infinite interval

Let Formula: see text be a positive, Lebesgue integrable and exponential decay function defined on an infinite interval Formula: see text and let Formula: see text be the space of weighted Lebesgue integrable functions on Formula: see text. In this paper, we give the relations of the best one-sided approximation and the optimal Hermite–Fejér interpolation by the set of algebraic polynomials of degree not exceeding a given number for the smooth function classes Formula: see text, Formula: see text, in the metric of the space Formula: see text and prove that the Hermite–Fejér interpolation based on the set of the zeros of some orthogonal polynomials is optimal in Formula: see text. In addition, we show that the approximation error of the optimal Hermite–Fejér interpolation and quadrature errors of the weighted Gaussian quadrature formula are equal, and give the exact constant of the errors.