Opuscula Mathematica (Feb 2024)

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

  • Soumitra Ghara,
  • Rajeev Gupta,
  • Md. Ramiz Reza

DOI
https://doi.org/10.7494/OpMath.2024.44.3.373
Journal volume & issue
Vol. 44, no. 3
pp. 373 – 390

Abstract

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

Keywords