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Mat. Zametki, 1992, Volume 52, Issue 3, Pages 123–129 (Mi mz4708)  

Equivalence of $K$-functional and modulus of smoothness of functions on the sphere

Kh. P. Rustamov

Institute of Applied Mathematics and Mechanics AS of AzSSR

Abstract: In the present note certain fundamental estimates of the constructive theory of functions on the sphere $S^n\subset\mathbf{R}^{n+1}$, $n\geqslant1$, are sharpened on the basis of the equivalence of the $K$-functional and the modulus of smoothness of functions. In particular a Bernshtein-type inequality for spherical polynomials is made more precise. The estimates obtained are applied to deduce a membership criterion for the function f $f\in L_p(S^n)$, $1\leqslant p\leqslant\infty$, to the space $H_r^{\omega}L_p(S^n)$ depending on the growth of the norm of derivatives of best approximation polynomials of the function $f$, which is an analog of a result found by S. B. Stechkin related to continuous periodic functions.

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English version:
Mathematical Notes, 1992, 52:3, 965–970

Bibliographic databases:

UDC: 517.5
Received: 04.03.1988

Citation: Kh. P. Rustamov, “Equivalence of $K$-functional and modulus of smoothness of functions on the sphere”, Mat. Zametki, 52:3 (1992), 123–129; Math. Notes, 52:3 (1992), 965–970

Citation in format AMSBIB
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\by Kh.~P.~Rustamov
\paper Equivalence of $K$-functional and modulus of smoothness of functions on the sphere
\jour Mat. Zametki
\yr 1992
\vol 52
\issue 3
\pages 123--129
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\transl
\jour Math. Notes
\yr 1992
\vol 52
\issue 3
\pages 965--970
\crossref{https://doi.org/10.1007/BF01209618}
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