Mean convergence of orthogonal Fourier series and interpolating polynomials

Abstract

For a family of weight functions that include the general Jacobi weight functions as special cases, exact condition for the convergence of the Fourier orthogonal series in the weighted L p space is given. The result is then used to establish a Marcinkiewicz-Zygmund type inequality and to study weighted mean convergence of various interpolating polynomials based on the zeros of the corresponding orthogonal polynomials.

Full Quadrature Sums for Generalized Polynomials With Freud Weights

“…A further generalization, valid for 0 < p < ∞, and P ∈ P ln with l > 1 fixed, was proved in [10]. A converse inequality has been proven in [16].…”

ON MARCINKIEWICZ-ZYGMUND INEQUALITIES IN LP,³-SPACES, 0 < p } 1

“…[1]. If n ≥ 1 is an integer, 1 < p < ∞, and S is a trigonometric polynomial of order at most n, then: These inequalities have been studied by many authors in connection with orthogonal polynomials both in the case of Lagrange interpolation and Jacobi weights on [ − 1 , 1 ] ( [2], [3]).…”