Polynomial Approximation in Higher-Order Weighted Dirichlet Spaces

TL;DR

This paper explores the failure of classical polynomial approximation in higher-order weighted Dirichlet spaces and introduces modified polynomials with weighted coefficients to achieve norm convergence.

Contribution

It develops a new summability method using modified polynomials to ensure convergence in higher-order weighted Dirichlet spaces, extending classical results.

Findings

Modified polynomials achieve norm convergence where classical sums diverge.

Explicit formulas for coefficients are derived for measures with Dirac masses.

A Fejér-type summability theorem is established for these spaces.

Abstract

Fej\'er's theorem guarantees norm convergence of Ces\`aro means of Taylor partial sums in the Hardy space, whereas such convergence generally fails in weighted Dirichlet-type spaces, especially in the higher-order setting. In this paper, we investigate summability problems in higher-order weighted Dirichlet spaces $\widehat{\mathcal{H}}_{\mu,m}$ and show that Taylor partial sums are not uniformly bounded in these spaces and may therefore diverge in norm. To restore convergence, we introduce a family of modified polynomials whose coefficients are adjusted by a suitable weight array. Under mild boundedness and variation assumptions on the weights, we establish norm convergence of the modified sums via a coefficient correspondence principle and a Local Douglas formula. As an application, when the weight measure $\mu$ is a finite sum of Dirac point masses, explicit formulas for the modified coefficients are obtained, yielding a Fej\'er-type summability theorem for higher-order weighted Dirichlet spaces.