Mean convergence of entire interpolations in weighted space

TL;DR

This paper studies the convergence behavior of entire Lagrange and Hermite interpolations of exponential type in weighted Lp spaces, providing new insights into their convergence properties based on weight functions.

Contribution

It introduces new convergence results for entire interpolations in weighted spaces, extending previous work to more general weight functions and types.

Findings

Convergence of interpolations is established under certain weight conditions.

Results generalize known theorems for power weights.

Marcinkiewicz inequalities are key to the analysis.

Abstract

We investigate the convergence of entire Lagrange interpolations and of Hermite interpolations of exponential type in weighted $L^p$-spaces on the real line. The weights are reciprocals of entire functions and depend on the type and may be viewed as smoothed versions of a target weight. The convergence statements are obtained from Marcinkiewicz inequalities with constants proportional to the type. For the special case of power weights we recover results of Rahman and Grozev and of Lubinsky.