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On strong means of spherical Fourier sums

Authors: O. I. Kuznetsova and A. N. Podkorytov
Translated by: N. V. Tsilevich
Original publication: Algebra i Analiz, tom 25 (2013), nomer 3.
Journal: St. Petersburg Math. J. 25 (2014), 447-453
MSC (2010): Primary 42B08
Published electronically: May 16, 2014
MathSciNet review: 3184600
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Abstract | References | Similar Articles | Additional Information

Abstract: The spherical Fourier sums

$\displaystyle S_n(f,x)=\sum _{\Vert k\Vert\leq n}\widehat f(k)\,e^{ik\cdot x} $

of a periodic function $ f$ in $ m$ variables and their strong means

$\displaystyle H_{n,p}(f,x)=\bigg (\frac 1n\sum _{j=0}^{n-1}\vert S_j(f,x)\vert^p\bigg )^{\frac 1p} \enskip$$\displaystyle \text {for}\enskip p\geq 1 $

are considered. In contrast to the one-dimensional case treated by Hardy and Littlewood, for $ m\geq 2$ the norms $ \sup _{\vert f\vert\leq 1}H_{n,p}(f,0)$ are not bounded. The sharp order of growth of these norms is found (the upper and lower bounds differ by a factor depending only on the dimension $ m$).

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

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Additional Information

O. I. Kuznetsova
Affiliation: Institute of Applied Mathematics and Mechanics, National Academy of Science of Ukraine, Roza Luksemburg, 74, 83114 Donetsk, Ukraine

A. N. Podkorytov
Affiliation: St. Petersburg State University, Universitetsky prospekt 28, Peterhof, St. Petersburg 198504, Russia

Keywords: Multiple Fourier series, spherical sums, strong means
Received by editor(s): October 5, 2012
Published electronically: May 16, 2014
Dedicated: To Boris Mikhaĭlovich Makarov
Article copyright: © Copyright 2014 American Mathematical Society