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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Cesàro summability of the conjugate series and the double Hilbert transform


Author: John O. Basinger
Journal: Proc. Amer. Math. Soc. 56 (1976), 177-182
MSC: Primary 42A40
DOI: https://doi.org/10.1090/S0002-9939-1976-0410231-9
MathSciNet review: 0410231
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Abstract: If $ f(x,y)$, a $ 2\pi $ periodic function in each variable, has a modulus of continuity $ {w_f}(\delta ) = o(1/\log (1/\delta ))$ then

$\displaystyle {\tilde \sigma _n}(x,y,f) - \int_{1/n}^\pi {\int_{1/n}^\pi {\frac... ...tan (u/2)\tan (v/2)}}} } dudv \to 0\quad {\text{uniformly}}\;{\text{in}}\;(x,y)$

where $ {\tilde \sigma _n}(x,y,f)$ is the first arithmetic mean of the conjugate series. This theorem is best possible in that $ o(1/\log (1/\delta ))$ cannot be replaced by $ O(1/\log (1/\delta ))$.

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DOI: https://doi.org/10.1090/S0002-9939-1976-0410231-9
Article copyright: © Copyright 1976 American Mathematical Society