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

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

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

Bibliographic databases:

UDC: 517.518.8
MSC: Primary 41A17; Secondary 22E30, 43A85

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

Citation in format AMSBIB
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• https://doi.org/10.4213/sm5096
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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.
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
3. S. S. Platonov, “Fourier–Jacobi harmonic analysis and approximation of functions”, Izv. Math., 78:1 (2014), 106–153
4. Victor S. Barbosa, Valdir A. Menegatto, “Generalized Convolution Roots of Positive Definite Kernels on Complex Spheres”, SIGMA, 11 (2015), 014, 13 pp.
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
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
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
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
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
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