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Izv. Akad. Nauk SSSR Ser. Mat., 1973, Volume 37, Issue 2, Pages 344–355 (Mi izv2251)  

This article is cited in 12 scientific papers (total in 12 papers)

On an extremal problem for polynomials in $n$ variables

Yu. A. Brudnyi, M. I. Ganzburg


Abstract: This article is devoted to an examination of the following extremal problem: find the quantity
$$ C_{k,n}(\lambda,B)=\sup_{|\omega|\ge\lambda}\sup_{P\in\mathscr P_{k,n}(\omega)}\|P\|_{C(B)}, $$
where $B$ is an $n$-dimensional sphere and $\mathscr P_{k,n}(\omega)$ is the totality of polynomials $P$ of degree $k$ in $n$ variables for which $\|P\|_{C(\omega)}\le1$. Here $\omega$ is a measurable set from $B$ and the first sup is taken over all measurable $\omega\subset B$ having measure $|\omega|\ge\lambda$.
The exact order of growth of $C_{k,n}(\lambda, B)$ which respect to $\lambda$ as $\lambda\to0$ is found in this article. Various applications of the results are examined as well.

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English version:
Mathematics of the USSR-Izvestiya, 1973, 7:2, 345–356

Bibliographic databases:

UDC: 577.514
MSC: Primary 41A10, 41A63; Secondary 41A50
Received: 31.05.1971

Citation: Yu. A. Brudnyi, M. I. Ganzburg, “On an extremal problem for polynomials in $n$ variables”, Izv. Akad. Nauk SSSR Ser. Mat., 37:2 (1973), 344–355; Math. USSR-Izv., 7:2 (1973), 345–356

Citation in format AMSBIB
\Bibitem{BruGan73}
\by Yu.~A.~Brudnyi, M.~I.~Ganzburg
\paper On an extremal problem for polynomials in~$n$ variables
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1973
\vol 37
\issue 2
\pages 344--355
\mathnet{http://mi.mathnet.ru/izv2251}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=352825}
\zmath{https://zbmath.org/?q=an:0283.26012}
\transl
\jour Math. USSR-Izv.
\yr 1973
\vol 7
\issue 2
\pages 345--356
\crossref{https://doi.org/10.1070/IM1973v007n02ABEH001941}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Evsey Dyn’kin, “Cauchy integral decomposition for harmonic forms”, J Anal Math, 73:1 (1997), 165  crossref  mathscinet  zmath  isi
    2. Y. Sagher, P. Shvartsman, “On the John-Strömberg-Torchinsky characterization of BMO”, The Journal of Fourier Analysis and Applications, 4:4-5 (1998), 521  crossref  mathscinet  zmath
    3. Proc. Steklov Inst. Math., 255 (2006), 65–81  mathnet  crossref  mathscinet
    4. I. P. Irodova, “O neravenstve tipa neravenstva Bernshteina”, Model. i analiz inform. sistem, 15:4 (2008), 31–41  mathnet  elib
    5. I. P. Irodova, “O neravenstve tipa neravenstva Dzheksona v diadicheskom prostranstve $BMO$”, Model. i analiz inform. sistem, 16:3 (2009), 29–46  mathnet  elib
    6. A. Brudnyi, Yu. Brudnyi, “Traces of $C^k$ functions to weak Markov subsets of $\mathbb R^n$”, St. Petersburg Math. J., 23:1 (2012), 39–56  mathnet  crossref  mathscinet  zmath  isi  elib
    7. A. I. Parfenov, “A characterization of multipliers in the Hedberg–Netrusov spaces”, Siberian Adv. Math., 22:1 (2012), 13–40  mathnet  crossref  mathscinet  elib
    8. I. P. Irodova, “Piecewise polynomial approximation methods in the theory of Nikol'skiĭ–Besov spaces”, Journal of Mathematical Sciences, 209:3 (2015), 319–480  mathnet  crossref
    9. Menny Aka, Emmanuel Breuillard, Lior Rosenzweig, Nicolas de Saxcé, “Diophantine properties of nilpotent Lie groups”, Compositio Math, 2015, 1  crossref
    10. Rafał Pierzchała, “Remez-type inequality on sets with cusps”, Advances in Mathematics, 281 (2015), 508  crossref
    11. V. I. Bogachev, “Distributions of polynomials on multidimensional and infinite-dimensional spaces with measures”, Russian Math. Surveys, 71:4 (2016), 703–749  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    12. Goldman G., “A Case of Multivariate Birkhoff Interpolation Using High Order Derivatives”, J. Approx. Theory, 223 (2017), 19–28  crossref  isi
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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