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Mat. Zametki, 1996, Volume 60, Issue 3, Pages 333–355 (Mi mz1834)  

This article is cited in 13 scientific papers (total in 13 papers)

Sharp Jackson–Stechkin inequality in $L^2$ for multidimensional spheres

A. G. Babenko

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: In this paper we prove the Jackson–Stechkin inequality
$$ E_{n-1}(f)<\omega_r(f,2\tau_{n,\lambda}), \qquad n\ge1, \quad m\ge5, \quad r\ge1, $$
$f\in L^2(\mathbb S^{m-1})$, $f\not\equiv\textrm{const}$, which is sharp for each $n=2,3,…$; here $E_{n-1}(f)$ is the best approximation of a function $f$ by spherical polynomials of degree $\le n-1$, $\omega_r(f,\tau)$ is the $r$th modulus of continuity of $f$ based on the translations
$$ s_tf(x)=\frac 1{|\mathbb S^{m-2}|}\int_{\mathbb S^{m-2}}f(x\cos t+\xi\sin t) d\xi, \qquad t\in\mathbb R, \quad x\in\mathbb S^{m-1}, $$
$\mathbb S^{m-2}=\mathbb S^{m-2}_x=\{\xi\in \mathbb S^{m-1}:x\cdot\xi=0\}$, $|\mathbb S^{m-2}|$ is the measure of the unit Euclidean sphere $\mathbb S^{m-2}$, $\lambda=(m-2)/2$ and $\tau_{n,\lambda}$ is the first positive zero of the Gegenbauer cosine polynomial $C^\lambda_n(\cos t)$ .


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English version:
Mathematical Notes, 1996, 60:3, 248–263

Bibliographic databases:

UDC: 517.518.837
Received: 04.04.1994
Revised: 18.06.1996

Citation: A. G. Babenko, “Sharp Jackson–Stechkin inequality in $L^2$ for multidimensional spheres”, Mat. Zametki, 60:3 (1996), 333–355; Math. Notes, 60:3 (1996), 248–263

Citation in format AMSBIB
\by A.~G.~Babenko
\paper Sharp Jackson--Stechkin inequality in $L^2$ for multidimensional spheres
\jour Mat. Zametki
\yr 1996
\vol 60
\issue 3
\pages 333--355
\jour Math. Notes
\yr 1996
\vol 60
\issue 3
\pages 248--263

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    This publication is cited in the following articles:
    1. V. I. Ivanov, O. I. Smirnov, “On Jackson's theorem in the space $\ell_2(\mathbb Z_2^n)$”, Math. Notes, 60:3 (1996), 288–299  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. A. G. Babenko, “An exact Jackson–Stechkin inequality for $L^2$-approximation on the interval with the Jacobi weight and on projective spaces”, Izv. Math., 62:6 (1998), 1095–1119  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. A. G. Babenko, N. I. Chernykh, V. T. Shevaldin, “The Jackson–Stechkin inequality in $L^2$ with a trigonometric modulus of continuity”, Math. Notes, 65:6 (1999), 777–781  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. D. V. Gorbachev, “The sharp Jackson inequality in the space $L_p$ on the sphere”, Math. Notes, 66:1 (1999), 40–50  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. E. E. Berdysheva, “Two related extremal problems for entire functions of several variables”, Math. Notes, 66:3 (1999), 271–282  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. V. A. Yudin, “Distribution of the points of a design on the sphere”, Izv. Math., 69:5 (2005), 1061–1079  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    7. Li J., Liu Y., “The Jackson Inequality for the Best L-2-Approximation of Functions on [0,1] with the Weight x”, Numerical Mathematics-Theory Methods and Applications, 1:3 (2008), 340–356  mathscinet  zmath  isi
    8. S. N. Vasil'ev, “Jackson inequality in $L_2(\mathbb T^N)$ with generalized modulus of continuity”, Proc. Steklov Inst. Math. (Suppl.), 265, suppl. 1 (2009), S218–S226  mathnet  crossref  isi  elib
    9. Ivanov V.I., “Tochnye $l_2$-neravenstva dzheksona - chernykh - yudina v teorii priblizhenii”, Izvestiya tulskogo gosudarstvennogo universiteta. estestvennye nauki, 2012, no. 3, 19–28  mathscinet  elib
    10. Liu Y.P., Song Ch.Yu., “Dunkl's Theory and Best Approximation By Entire Functions of Exponential Type in $L_2$-Metric With Power Weight”, Acta. Math. Sin.-English Ser., 30:10 (2014), 1748–1762  crossref  mathscinet  zmath  isi  scopus  scopus
    11. Gu Y., Liu Y., “The Sharp Jackson Inequality For l-2-Approximation on the Periodic Cylinder”, Acta Math. Sci., 35:2 (2015), 375–382  crossref  mathscinet  zmath  adsnasa  isi  scopus
    12. R. A. Lasuriya, “Direct and Inverse Theorems on the Approximation of Functions by Fourier–Laplace Sums in the Spaces $S^{(p,q)}(\sigma^{m-1})$”, Math. Notes, 98:4 (2015), 601–612  mathnet  crossref  crossref  mathscinet  isi  elib
    13. R. A. Lasuriya, “Jackson-Type Inequalities in the Spaces $S^{(p,q)}(\sigma^{m-1})$”, Math. Notes, 105:5 (2019), 707–719  mathnet  crossref  crossref  isi  elib
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