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Mat. Sb., 2009, Volume 200, Number 6, Pages 67–108 (Mi msb5096)  

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

Some problems in the theory of approximation of functions on compact homogeneous manifolds

S. S. Platonov

Petrozavodsk State University, Faculty of Mathematics

Abstract: Problems in the theory of approximation of functions on an arbitrary compact rank-one symmetric space $M$ in the metric of $L_p$, $1\le p\le\infty$, are investigated. The approximating functions are generalized spherical polynomials, that is, linear combinations of eigenfunctions of the Beltrami-Laplace operator on $M$. Analogues of the direct Jackson theorems are proved for the modulus of smoothness (of arbitrary order) constructed by using the operator of spherical averaging. It is established that the modulus of smoothness and the $K$-functional constructed from the Sobolev-type space corresponding to the Beltrami-Laplace differential operator are equivalent.
Bibliography: 35 titles.

Keywords: approximation of functions, compact symmetric space, Jacobi polynomials, moduli of smoothness, Jackson's theorems.

DOI: https://doi.org/10.4213/sm5096

Full text: PDF file (760 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2009, 200:6, 845–885

Bibliographic databases:

UDC: 517.518.8
MSC: Primary 41A17; Secondary 22E30, 43A85
Received: 28.03.2008 and 09.12.2008

Citation: S. S. Platonov, “Some problems in the theory of approximation of functions on compact homogeneous manifolds”, Mat. Sb., 200:6 (2009), 67–108; Sb. Math., 200:6 (2009), 845–885

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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. El Ouadih S., Daher R., “On Spherical Analogues of the Classical Theorems of Titchmarsh”, Integral Transform. Spec. Funct.  crossref  mathscinet  isi
    2. El Ouadih S., Daher R., Tyr O., Saadi F., “Equivalence of K-Functionals and Moduli of Smoothness Generated By the Beltrami-Laplace Operator on the Spaces S-(P,S-Q)(SIGMA(M-1))”, Rend. Circ. Mat. Palermo  crossref  isi
    3. S. S. Platonov, “Fourier–Jacobi harmonic analysis and approximation of functions”, Izv. Math., 78:1 (2014), 106–153  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. Victor S. Barbosa, Valdir A. Menegatto, “Generalized Convolution Roots of Positive Definite Kernels on Complex Spheres”, SIGMA, 11 (2015), 014, 13 pp.  mathnet  crossref  mathscinet
    5. Jordao T., Menegatto V.A., “Jackson kernels: a tool for analysing the decay of eigenvalue sequences of integral operators on the sphere”, Math. Inequal. Appl., 18:4 (2015), 1483–1500  crossref  mathscinet  zmath  isi  scopus
    6. Barbosa V.S., Menegatto V.A., “Strictly positive definite kernels on compact two-point homogeneous spaces”, Math. Inequal. Appl., 19:2 (2016), 743–756  crossref  mathscinet  zmath  isi  scopus
    7. Bonfim R.N., Menegatto V.A., “Strict positive definiteness of multivariate covariance functions on compact two-point homogeneous spaces”, J. Multivar. Anal., 152 (2016), 237–248  crossref  mathscinet  zmath  isi  scopus
    8. Castro M.H., Jordao T., Peron A.P., “Super-Exponential Decay Rates For Eigenvalues and Singular Values of Integral Operators on the Sphere”, J. Comput. Appl. Math., 364 (2020), UNSP 112334  crossref  mathscinet  isi
    9. Carrijo A.O., Jordao T., “Approximation Tools and Decay Rates For Eigenvalues of Integral Operators on a General Setting”, Positivity, 24:4 (2020), 761–777  crossref  mathscinet  isi
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