(C, α) summability of Walsh-Fourier series

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The one-dimensional dyadic martingale Hardy spaces Hp are introduced and it is proved that the maximal operator of the (C, α) means of a Walsh-Fourier series is bounded from Hp to Lp (1/(α + 1) < p < ∞) and is of weak type (L1, L1). As a consequence, we obtain the summability result due to Fine; more exactly, the (C, α) means of the Walsh-Fourier series of a function f ∈ L1 converge a.e. to the function in question. Moreover, we prove that the (C, α) means are uniformly bounded on Hp whenever 1/(α + 1) < p < ∞. We define the two-dimensional dyadic hybrid Hardy space H1# and verify that the maximal operator of the (C, α, β) means of a two-dimensional function is of weak type (H1#, L1). Consequence, the Walsh-Fourier series of every function of f ε H1# is (C, α, β) summable to the function f.

Original languageEnglish
Pages (from-to)141-155
Number of pages15
JournalAnalysis Mathematica
Issue number2
Publication statusPublished - Jan 1 2001

ASJC Scopus subject areas

  • Mathematics(all)

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