Mean convergence of orthogonal Fourier series and interpolating polynomials

TL;DR

The paper addresses the weighted mean convergence of Fourier orthogonal series and the mean convergence of interpolating polynomials when the underlying weights belong to the generalized Jacobi class. It develops a double-weight Hilbert transform inequality for GJ weights, derives precise convergence criteria for the partial sum operators $S_n(d\alpha,f)$ in $L^p(d\beta)$, and establishes Marcinkiewicz-Zygmund inequalities for polynomials evaluated at the zeros of orthogonal polynomials. These results are extended to Hermite interpolation and higher-order interpolation schemes, providing a unified framework for weighted approximation with broad weight classes, including $GJ\log$ weights. The methods combine orthogonal polynomial theory, kernel estimates, and weighted harmonic analysis to obtain both necessary and sufficient conditions for convergence and quantitative bounds for interpolants, enhancing the toolkit for weighted approximation in numerical analysis and related fields.

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.

Theorems & Definitions (37)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8
• Lemma 3.1
• Lemma 3.2