Opuscula Math. 44, no. 3 (2024), 373-390
https://doi.org/10.7494/OpMath.2024.44.3.373

 
Opuscula Mathematica

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

Soumitra Ghara
Rajeev Gupta
Md. Ramiz Reza

Abstract. 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: weighted Dirichlet-type integrals, Cesàro mean, summability, Hadamard multiplication.

Mathematics Subject Classification: 41A10, 40G05, 46E20, 41A17.

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  • Soumitra Ghara
  • Department of Mathematics, Indian Institute of Technology Kharagpur, Midnapore - 721302, India
  • Rajeev Gupta
  • School of Mathematics and Computer Science, Indian Institute of Technology Goa, Goa - 403401, India
  • Md. Ramiz Reza (corresponding author)
  • School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, Kerala - 695551, India
  • Communicated by P.A. Cojuhari.
  • Received: 2024-01-21.
  • Revised: 2024-02-04.
  • Accepted: 2024-02-07.
  • Published online: 2024-02-15.
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Cite this article as:
Soumitra Ghara, Rajeev Gupta, Md. Ramiz Reza, Cesàro summability of Taylor series in higher order weighted Dirichlet-type spaces, Opuscula Math. 44, no. 3 (2024), 373-390, https://doi.org/10.7494/OpMath.2024.44.3.373

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