Some new results on H summability of Fourier series

In this paper we shall be concerned with H α summability, for 0 < α ≤ 2 of the Fourier series of arbitrary L 1 ([−π, π]) functions. The methods employed here are a modification of the real variable ones introduced by J. Marcinkiewicz. The needed modifications give direct proofs of maximal theorem...

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Bibliographic Details
Published in:Quaestiones mathematicae 2019-10, Vol.42 (8), p.1045-1064
Main Authors: Calderón, Calixto P., Coré, A. Susana, Urbina, Wilfredo O.
Format: Article
Language:English
Subjects:
Density
Fourier series
Marcinkiewicz function
Primary 42B08
Real variables
Secondary 26A24
strong summability
Theorems
weights
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Summary:In this paper we shall be concerned with H α summability, for 0 < α ≤ 2 of the Fourier series of arbitrary L 1 ([−π, π]) functions. The methods employed here are a modification of the real variable ones introduced by J. Marcinkiewicz. The needed modifications give direct proofs of maximal theorems with respect to A 1 weights. We also give a counter-example of a measure such that there is no convergence a.e. to the density of the measure. Finally, we present a Kakutani type of theorem, proving the ω*-density, in the space of of probability measures defined on [−π, π] of Borel measures for which there is no H 2 summability a.e.
ISSN:1607-3606
1727-933X
DOI:10.2989/16073606.2018.1504253