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Mat. Zametki, 2004, Volume 76, Issue 5, Pages 723–731 (Mi mz139)  

This article is cited in 7 scientific papers (total in 7 papers)

Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series

M. I. Dyachenko

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: It follows from results of A. Yudin, V. Yudin, E. Belinskii, and I. Liflyand that if $m\ge2$ and a $2\pi$-periodic (in each variable) function $f(\mathbf x)\in C(T^m)$ belongs to the Nikol'skii class $h_\infty^{(m-1)/2}(T^m)$, then its multiple Fourier series is uniformly convergent over hyperbolic crosses. In this paper, we establish the finality of this result. More precisely, there exists a function in the class $h_\infty^{(m-1)/2}(T^m)$ whose Fourier series is divergent over hyperbolic crosses at some point.

DOI: https://doi.org/10.4213/mzm139

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English version:
Mathematical Notes, 2004, 76:5, 673–681

Bibliographic databases:

UDC: 517.51.475
Received: 01.10.2003

Citation: M. I. Dyachenko, “Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series”, Mat. Zametki, 76:5 (2004), 723–731; Math. Notes, 76:5 (2004), 673–681

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. Yu. Trynin, “On necessary and sufficient conditions for convergence of sinc-approximations”, St. Petersburg Math. J., 27:5 (2016), 825–840  mathnet  crossref  mathscinet  isi  elib
    2. A. Yu. Trynin, “On some properties of sinc approximations of continuous functions on the interval”, Ufa Math. J., 7:4 (2015), 111–126  mathnet  crossref  isi  elib
    3. A. Yu. Trynin, “Approximation of continuous on a segment functions with the help of linear combinations of sincs”, Russian Math. (Iz. VUZ), 60:3 (2016), 63–71  mathnet  crossref  isi
    4. A. Yu. Trynin, “Uniform convergence of Lagrange–Sturm–Liouville processes on one functional class”, Ufa Math. J., 10:2 (2018), 93–108  mathnet  crossref  isi
    5. A. Yu. Trynin, “Skhodimost protsessov Lagranzha–Shturma–Liuvillya dlya nepreryvnykh funktsii ogranichennoi variatsii”, Vladikavk. matem. zhurn., 20:4 (2018), 76–91  mathnet  crossref  elib
    6. A. Yu. Trynin, “Sufficient condition for convergence of Lagrange–Sturm–Liouville processes in terms of one-sided modulus of continuity”, Comput. Math. Math. Phys., 58:11 (2018), 1716–1727  mathnet  crossref  crossref  isi  elib
    7. A. Yu. Trynin, E. D. Kireeva, “Printsip lokalizatsii na klasse funktsii, integriruemykh po Rimanu, dlya protsessov Lagranzha–Shturma–Liuvillya”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 20:1 (2020), 51–63  mathnet  crossref
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