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Mat. Zametki, 2008, Volume 84, Issue 4, Pages 532–551 (Mi mz6137)  

This article is cited in 9 scientific papers (total in 9 papers)

The Taikov Functional in the Space of Algebraic Polynomials on the Multidimensional Euclidean Sphere

M. V. Deikalova

Ural State University

Abstract: We discuss three related extremal problems on the set $\mathscr P_{n,m}$ of algebraic polynomials of given degree $n$ on the unit sphere $\mathbb S^{m-1}$ of Euclidean space $\mathbb R^m$ of dimension $m\ge 2$. (1) The norm of the functional $F(h)=F_hP_n=\int_{\mathbb C(h)}P_n(x) dx$, which is equal to the integral over the spherical cap $\mathbb C(h)$ of angular radius $\operatorname{arccos} h$, $-1<h<1$, on the set $\mathscr P_{n,m}$ with the norm of the space $L(\mathbb S^{m-1})$ of summable functions on the sphere. (2) The best approximation in $L_\infty(\mathbb S^{m-1})$ of the characteristic function $\chi_h$ of the cap $\mathbb C(h)$ by the subspace $\mathscr P^\bot_{n,m}$ of functions from $L_\infty(\mathbb S^{m-1})$ that are orthogonal to the space of polynomials $\mathscr P_{n,m}$. (3) The best approximation in the space $L(\mathbb S^{m-1})$ of the function $\chi_h$ by the space of polynomials $\mathscr P_{n,m}$. We present the solution of all three problems for the value $h=t(n,m)$ which is the largest root of the polynomial in a single variable of degree $n+1$ least deviating from zero in the space $L_1^\phi$ on the interval $(-1,1)$ with ultraspheric weight $\phi(t)=(1-t^2)^\alpha$, $\alpha=(m-3)/2$.

Keywords: Taikov functional, algebraic polynomial, Euclidean sphere, spherical cap, polynomial of least deviation, Lebesgue measure, Hahn–Banach theorem, zonal function

DOI: https://doi.org/10.4213/mzm6137

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English version:
Mathematical Notes, 2008, 84:4, 498–514

Bibliographic databases:

UDC: 517.518.86
Received: 31.12.2007
Revised: 11.01.2008

Citation: M. V. Deikalova, “The Taikov Functional in the Space of Algebraic Polynomials on the Multidimensional Euclidean Sphere”, Mat. Zametki, 84:4 (2008), 532–551; Math. Notes, 84:4 (2008), 498–514

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. G. Babenko, Yu. V. Kryakin, “Integral approximation of the characteristic function of an interval by trigonometric polynomials”, Proc. Steklov Inst. Math. (Suppl.), 264, suppl. 1 (2009), S19–S38  mathnet  crossref  isi  elib
    2. M. V. Deikalova, “About the sharp Jackson–Nikol'skii inequality for algebraic polynomials on a multidimensional Euclidean sphere”, Proc. Steklov Inst. Math. (Suppl.), 265, suppl. 1 (2009), S129–S142  mathnet  crossref  isi  elib
    3. M. V. Deikalova, “The integral approximation of the characteristic function of a spherical cap by algebraic polynomials”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S74–S85  mathnet  crossref  isi  elib
    4. M. V. Deikalova, “Several extremal approximation problems for the characteristic function of a spherical layer”, Proc. Steklov Inst. Math. (Suppl.), 277, suppl. 1 (2012), 79–92  mathnet  crossref  isi  elib
    5. A. V. Efimov, “A version of the Turan problem for positive definite functions of several variables”, Proc. Steklov Inst. Math. (Suppl.), 277, suppl. 1 (2012), 93–112  mathnet  crossref  isi  elib
    6. M. V. Deikalova, V. V. Rogozina, “Neravenstvo Dzheksona–Nikolskogo mezhdu ravnomernoi i integralnoi normami algebraicheskikh mnogochlenov na evklidovoi sfere”, Tr. IMM UrO RAN, 18, no. 4, 2012, 162–171  mathnet  elib
    7. V. V. Arestov, M. V. Deikalova, “Nikol'skii inequality for algebraic polynomials on a multidimensional Euclidean sphere”, Proc. Steklov Inst. Math. (Suppl.), 284, suppl. 1 (2014), 9–23  mathnet  crossref  mathscinet  isi  elib
    8. Marina V. Deikalova, Anastasiya Yu. Torgashova, “One-sided $L$-approximation on a sphere of the characteristic function of a layer”, Ural Math. J., 4:2 (2018), 13–23  mathnet  crossref  mathscinet
    9. V. V. Arestov, A. A. Seleznev, “Nailuchshee $L^2$-prodolzhenie algebraicheskikh mnogochlenov s edinichnoi evklidovoi sfery na kontsentricheskuyu sferu”, Tr. IMM UrO RAN, 26, no. 2, 2020, 47–55  mathnet  crossref  elib
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