Fejér summability of multi-parameter bounded Ciesielski systems

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Abstract

We investigate the Kronecker product of bounded Ciesielski systems, which can be obtained from the spline systems of order (m, k) in the same way as the Walsh system from the Haar system. It is shown that the maximal operator of the Fejér means of the d-dimensional Ciesielski-Fourier series is bounded from the Hardy space Hp([0, 1)d1×...×[0,1)dl to Lp ([0, 1)d) if 1/2 < p < ∞ and mj ≥0, {pipe}kj{pipe} ≤ mj+1. By an interpolation theorem, we get that the maximal operator is also of weak type H1#i, L1 (i=1,...,l), where the Hardy space H1#i is defined by a hybrid maximal function and H1#i ⊃ L(log L)l-1. As a consequence, we obtain that the Fejér means of the Ciesielski-Fourier series of a function f converge to f a.e. if f ∈ H1#i and converge in a cone if fεL1.

Original languageEnglish
Pages (from-to)135-155
Number of pages21
JournalAnalysis Mathematica
Volume28
Issue number2
DOIs
Publication statusPublished - Jan 1 2002

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ASJC Scopus subject areas

  • Mathematics(all)

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