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A note on the strong summability of the Riesz means of multiple Fourier series


Author: Shigehiko Kuratsubo
Journal: Proc. Amer. Math. Soc. 96 (1986), 294-298
MSC: Primary 42A28; Secondary 42B05
DOI: https://doi.org/10.1090/S0002-9939-1986-0818461-5
MathSciNet review: 818461
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Abstract: Let $ S_t^\alpha (x,f)$ be the Riesz means of order $ \alpha $ of an integrable function $ f(x)$ on $ N$-dimensional torus $ {T^N}(N \geqslant 2)$, that is,

$\displaystyle S_t^\alpha (x,f) = \sum\limits_{{{\left\vert m \right\vert}^2} < ... ...{\left\vert m \right\vert}^2}}}{t}} \right)}^\alpha }\hat f(m){e^{2\pi imx}}.} $

E. M. Stein has shown that if $ 1 < p \leqslant 2$ and $ \alpha > {\alpha _p}$ where

$\displaystyle {\alpha _p} = \frac{{N - 1}}{2}\left( {\frac{2}{p} - 1} \right) - \frac{1}{{p'}} = \frac{{N - 1}}{2} - \frac{N}{{p'}},$

then for any function $ f(x) \in {L^p}({T^N})S_t^\alpha (x,f)$ is strong summable to $ f(x)$, that is,

$\displaystyle \mathop {\lim }\limits_{T \to \infty } \frac{1}{T}\int_0^T {{{\left\vert {S_t^\alpha (x,f) - f(x)} \right\vert}^2}dt = 0} $

for almost every $ x$. In this paper we shall show that if $ 1 \leqslant p \leqslant 2$ and $ - 1 < \alpha < {\alpha _p}$, then there exists a function $ f(x) \in {L^p}({T^N})$ such that

$\displaystyle \frac{1}{T} \int_0^T \vert S_t^\alpha(x,f)\vert^2 dt - \Omega(T^{\alpha_p - \alpha} \log^{-2\tau} T)$   as $ T \to \infty$$\displaystyle $

for every $ x$ and every $ \tau > 1/p$, in particular,

$\displaystyle \mathop{\overline{\lim}}\limits_{T \to \infty } \frac{1}{T} \int_0^T \vert S_t^\alpha(x,f) \vert^2 dt = \infty $

for every $ x$, where we can take for $ f(x),{f_{\sigma \tau }}(x)$ such that $ {\hat f_{\sigma \tau }}(m) = 1/{\left\vert m \right\vert^\sigma }{\log ^\tau }\left\vert m \right\vert,\left\vert m \right\vert > 1$.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0818461-5
Keywords: Multiple Fourier series, Riesz means, strong summability, $ {L^p}({T^N})$, lattice point problem
Article copyright: © Copyright 1986 American Mathematical Society