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Cesàro summability of Taylor series in higher order weighted Dirichlet type spaces

Soumitra Ghara, Rajeev Gupta, Md. Ramiz Reza

TL;DR

This work analyzes Cesàro summability of Taylor series in higher-order weighted Dirichlet-type spaces $\mathcal{H}_{\mu,m}$. By extending Hadamard multiplier theory to these spaces and employing reproducing kernel Hilbert space techniques, it proves that generalized Cesàro means $\sigma_n^{\alpha}[f]$ converge to $f$ in $\mathcal{H}_{\mu,m}$ for all measures $\mu$ when $\alpha>\frac{1}{2}$, and provides a norm-convergence guarantee in $\|\cdot\|_{\mu,m}$. The paper also shows that the boundary case $\alpha=\frac{1}{2}$ fails in local spaces $\mathcal{H}_{\lambda,m}$ via counterexamples, highlighting a sharp threshold for summability. An alternative induction-based proof using the backward-shift operator complements the main approach, offering a second route to the same convergence result and enriching the theory of $m$-isometries in analytic function spaces.

Abstract

For a positive integer $m$ and a finite non-negative Borel measure $μ$ on the unit circle, we study the Hadamard multipliers of higher order weighted Dirichlet-type spaces $\mathcal H_{μ, m}$. We show that if $α>\frac{1}{2},$ then for any $f$ in $\mathcal H_{μ, m},$ the sequence of generalized Ces{à}ro sums $\{σ_n^α[f]\}$ converges to $f$. We further show that if $α=\frac{1}{2}$ then for the Dirac delta measure supported at any point on the unit circle, the previous statement breaks down for every positive integer $m$.

Cesàro summability of Taylor series in higher order weighted Dirichlet type spaces

TL;DR

This work analyzes Cesàro summability of Taylor series in higher-order weighted Dirichlet-type spaces . By extending Hadamard multiplier theory to these spaces and employing reproducing kernel Hilbert space techniques, it proves that generalized Cesàro means converge to in for all measures when , and provides a norm-convergence guarantee in . The paper also shows that the boundary case fails in local spaces via counterexamples, highlighting a sharp threshold for summability. An alternative induction-based proof using the backward-shift operator complements the main approach, offering a second route to the same convergence result and enriching the theory of -isometries in analytic function spaces.

Abstract

For a positive integer and a finite non-negative Borel measure on the unit circle, we study the Hadamard multipliers of higher order weighted Dirichlet-type spaces . We show that if then for any in the sequence of generalized Ces{à}ro sums converges to . We further show that if then for the Dirac delta measure supported at any point on the unit circle, the previous statement breaks down for every positive integer .
Paper Structure (4 sections, 12 theorems, 69 equations)

This paper contains 4 sections, 12 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Hadamard Multiplication Operators on $\mathcal{H}_{\mu,m}$
  3. Generalized Cesàro summability in higher order Dirichlet spaces
  4. Convergence of Generalized Cesàro sum

Key Result

Theorem 1.1

Let $\mu\in\mathcal{M}_+(\mathbb T)$ and $m\in \mathbb N.$ If $\alpha > \frac{1}{2}$ then there exists a constant $\kappa >0$ such that Moreover, for every $f\in \mathcal{H}_{\mu,m},$$D_{\mu,m}(\sigma_n^{\alpha}[f]-f) \rightarrow 0$ as $n\to \infty.$

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • ...and 12 more