Constructive Approximation

, Volume 43, Issue 2, pp 291–309

On Summation of Nonharmonic Fourier Series

Article

DOI: 10.1007/s00365-015-9290-6

Cite this article as:
Belov, Y. & Lyubarskii, Y. Constr Approx (2016) 43: 291. doi:10.1007/s00365-015-9290-6

Abstract

Let a sequence \(\Lambda \subset {\mathbb {C}}\) be such that the corresponding system of exponential functions \({\mathcal {E}}(\Lambda ):=\left\{ {\text {e}}^{i\lambda t}\right\} _{\lambda \in \Lambda }\) is complete and minimal in \(L^2(-\pi ,\pi )\), and thus each function \(f\in L^2(-\pi ,\pi )\) corresponds to a nonharmonic Fourier series in \({\mathcal {E}}(\Lambda )\). We prove that if the generating function \(G\) of \(\Lambda \) satisfies the Muckenhoupt \((A_2)\) condition on \({\mathbb {R}}\), then this series admits a linear summation method. Recent results show that the \((A_2)\) condition cannot be omitted.

Keywords

Nonharmonic Fourier series Summation methods  Paley–Wiener spaces Muckenhoupt condition Lagrange interpolation 

Mathematics Subject Classification

42A24 42A63 30D10 

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Chebyshev LaboratorySt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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