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- W2039234489 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, [ S_t^alpha (x,f) = sum limits _{{{left | m right |}^2} < t} {{{left ( {1 - frac {{{{left | m right |}^2}}}{t}} right )}^alpha }hat f(m){e^{2pi imx}}.} ] E. M. Stein has shown that if $1 < p leqslant 2$ and $alpha > {alpha _p}$ where [ {alpha _p} = frac {{N - 1}}{2}left ( {frac {2}{p} - 1} right ) - frac {1}{{pâ}} = frac {{N - 1}}{2} - frac {N}{{pâ}},] then for any function $f(x) in {L^p}({T^N})S_t^alpha (x,f)$ is strong summable to $f(x)$, that is, [ lim limits _{T to infty } frac {1}{T}int _0^T {{{left | {S_t^alpha (x,f) - f(x)} right |}^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 [ frac {1}{T} int _0^T |S_t^alpha (x,f)|^2 dt - Omega (T^{alpha _p - alpha } log ^{-2tau } T) quad text {as $T to infty $} ] for every $x$ and every $tau > 1/p$, in particular, [ overline {lim }limits _{T to infty } frac {1}{T} int _0^T | S_t^alpha (x,f) |^2 dt = infty ] for every $x$, where we can take for $f(x),{f_{sigma tau }}(x)$ such that ${hat f_{sigma tau }}(m) = 1/{left | m right |^sigma }{log ^tau }left | m right |,left | m right | > 1$." @default.
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- W2039234489 date "1986-02-01" @default.
- W2039234489 modified "2023-10-18" @default.
- W2039234489 title "A note on the strong summability of the Riesz means of multiple Fourier series" @default.
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- W2039234489 doi "https://doi.org/10.1090/s0002-9939-1986-0818461-5" @default.
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