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2 results

  • ON THE MINIMAL PROPERTY OF DE LA VALLÉE POUSSIN’S OPERATOR

  • Part of
  • BEATA DEREGOWSKA, BARBARA LEWANDOWSKA
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 91 / Issue 1 / February 2015
    Published online by Cambridge University Press:
    08 October 2014, pp. 129-133
    Print publication:
    February 2015
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  • Let $X={\mathcal{C}}_{0}(2{\it\pi})$ or $X=L_{1}[0,2{\it\pi}]$. Denote by ${\rm\Pi}_{n}$ the space of all trigonometric polynomials of degree less than or equal to $n$. The aim of this paper is to prove the minimality of the norm of de la Vallée Poussin’s operator in the set of generalised projections ${\mathcal{P}}_{{\rm\Pi}_{n}}(X,\,{\rm\Pi}_{2n-1})=\{P\in {\mathcal{L}}(X,{\rm\Pi}_{2n-1}):P|_{{\rm\Pi}_{n}}\equiv \text{id}\}$.

  • Walsh-Fourier series with coefficients of generalized bounded variation

  • Part of
  • F. Móricz
  • Journal:
    Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics / Volume 47 / Issue 3 / December 1989
    Published online by Cambridge University Press:
    09 April 2009, pp. 458-465
    Print publication:
    December 1989
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  • We extend in different ways the class of null sequences of real numbers that are of bounded variation and study the Walsh-Fourier series of integrable functions on the interval [(0, 1) with such coefficients. We prove almost everywhere convergence as well as convergence in the pseu dometric of Lr(0, 1) for 0 < r < 1.