Geodesic
Some problems in the theory of approximation of functions on compact homogeneous manifolds
Sbornik. Mathematics, Tome 200 (2009) no. 6, pp. 845-885 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{SM_2009_200_6_a2,
     author = {S. S. Platonov},
     title = {Some problems in the theory of approximation of functions on compact homogeneous manifolds},
     journal = {Sbornik. Mathematics},
     pages = {845--885},
     year = {2009},
     volume = {200},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2009_200_6_a2/}
}
TY  - JOUR
AU  - S. S. Platonov
TI  - Some problems in the theory of approximation of functions on compact homogeneous manifolds
JO  - Sbornik. Mathematics
PY  - 2009
SP  - 845
EP  - 885
VL  - 200
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/SM_2009_200_6_a2/
LA  - en
ID  - SM_2009_200_6_a2
ER  - 
%0 Journal Article
%A S. S. Platonov
%T Some problems in the theory of approximation of functions on compact homogeneous manifolds
%J Sbornik. Mathematics
%D 2009
%P 845-885
%V 200
%N 6
%U http://geodesic.mathdoc.fr/item/SM_2009_200_6_a2/
%G en
%F SM_2009_200_6_a2
S. S. Platonov. Some problems in the theory of approximation of functions on compact homogeneous manifolds. Sbornik. Mathematics, Tome 200 (2009) no. 6, pp. 845-885. http://geodesic.mathdoc.fr/item/SM_2009_200_6_a2/

[1] A. L. Besse, Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb., 93, Springer-Verlag, Berlin–Heidelberg–New York, 1978 | MR | MR | Zbl

[2] K. Wang, L. Q. Li, Harmonic analysis and approximation on the unit sphere, Science Press, Beijing, 2000

[3] Yu. Xu, “Weighted approximation of functions on the unit sphere”, Constr. Approx., 21:1 (2004), 1–28 | DOI | MR | Zbl

[4] Kh. P. Rustamov, “On approximation of functions on the sphere”, Russian Acad. Sci. Izv. Math., 43:2 (1994), 311–329 | DOI | MR | Zbl

[5] S. M. Nikolskii, “Approximation on manifolds”, East J. Approx., 1:1 (1995), 1–24 | MR | Zbl

[6] S. M. Nikol'skiǐ, P. I. Lizorkin, “Approximation of functions on the sphere”, Math. USSR-Izv., 30:3 (1988), 599–614 | DOI | MR | Zbl | Zbl

[7] D. L. Ragozin, “Constructive polynomial approximation on spheres and projective spaces”, Trans. Amer. Math. Soc., 162 (1971), 157–170 | DOI | MR | Zbl

[8] A. I. Kamzolov, “On Riesz's interpolational formula and Bernshtein's inequality for functions on homogeneous spaces”, Math. Notes, 15:6 (1974), 576–582 | DOI | MR | Zbl | Zbl

[9] A. I. Kamzolov, “Bohr–Favard inequality for functions on compact symmetric spaces of rank one”, Math. Notes, 33:2 (1983), 95–98 | DOI | MR | Zbl | Zbl

[10] V. A. Ivanov, “On the Bernstein–Nikol'skii and Favard inequalities on compact homogeneous spaces of rank 1”, Russian Math. Surveys, 38:3 (1983), 145–146 | DOI | MR | Zbl | Zbl

[11] V. M. Tikhomirov, “Teoriya priblizhenii”, Analiz – 2, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 14, VINITI, M., 1987, 103–260 | MR | Zbl

[12] L. Q. Li, “Riesz means on compact Riemannian symmetric spaces”, Math. Nachr., 168 (1994), 227–242 | MR | Zbl

[13] S. S. Platonov, “Approximations on compact symmetric spaces of rank 1”, Sb. Math., 188:5 (1997), 753–769 | DOI | MR | Zbl

[14] S. S. Platonov, “On equivalent normings in the spaces $H_p^r$ on compact homogeneous spaces”, Math. Notes, 68:5–6 (2000), 760–769 | DOI | MR | Zbl

[15] S. S. Platonov, “Jackson type theorems for compact rank 1 symmetric spaces”, Siberian Math. J., 42:1 (2001), 119–130 | DOI | MR | Zbl

[16] 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 | DOI | MR | Zbl

[17] S. Helgason, Differential geometry and symmetric spaces, Pure Appl. Math., 12, Academic Press, New York–London, 1962 | MR | Zbl | Zbl

[18] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure Appl. Math., 80, Academic Press, New York–San Francisco–London, 1978 | MR | Zbl

[19] G. A. Kalyabin, “On moduli of smoothness of functions given in the sphere”, Soviet Math. Dokl., 35:3 (1987), 619–622 | MR | Zbl

[20] S. S. Platonov, “Approximation of functions in the $L_2$-metric on noncompact symmetric spaces of rank 1”, St. Petersburg Math. J., 11:1 (2000), 183–201 | MR | Zbl

[21] 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 | DOI | MR | Zbl

[22] S. M. Nikol'skiĭ, Approximation of functions of several variables and imbedding theorems, Grundlehren Math. Wiss., 205, Springer-Verlag, New York–Heidelberg, 1975 | MR | MR | Zbl | Zbl

[23] H. Johnen, K. Scherer, “On the equivalence of the $K$-functional and moduli of continuity and some applications”, Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976), Lecture Notes in Math., 571, Springer-Verlag, Berlin, 1977, 119–140 | DOI | MR | Zbl

[24] R. A. DeVore, G. G. Lorentz, Constructive approximation, Grundlehren Math. Wiss., 303, Springer-Verlag, Berlin, 1993 | MR | Zbl

[25] S. Helgason, Groups and geometric analysis, Pure Appl. Math., 113, Academic Press, Orlando, FL, 1984 | MR | MR | Zbl

[26] S. Helgason, “The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds”, Acta Math., 113:1 (1965), 153–180 | DOI | MR | Zbl

[27] S. Helgason, Geometric analysis on symmetric spaces, Math. Surveys Monogr., 39, Amer. Math. Soc., Providence, RI, 1994 | MR | Zbl

[28] Th. Q. Sherman, “The Helgason Fourier transform for compact Riemannian symmetric spaces of rank one”, Acta Math., 164:1 (1990), 73–144 | DOI | MR | Zbl

[29] G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., 23, Amer. Math. Soc., Providence, RI, 1959 | MR | Zbl | Zbl

[30] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, vols. I, II, McGraw-Hill, New York–Toronto–London, 1953 | MR | MR | MR | Zbl | Zbl

[31] I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, Academic Press, New York–London–Toronto, ON, 1980 | MR | MR | Zbl | Zbl

[32] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of integral transforms, vols. I, II, McGraw-Hill, New York–Toronto–London, 1954 | MR | MR | MR | Zbl | Zbl

[33] S. M. Nikolskii, “Obobschenie odnogo neravenstva S. N. Bernshteina”, Dokl. AN SSSR, 60:9 (1948), 1507–1510 | MR | Zbl

[34] S. B. Stechkin, “Obobschenie nekotorykh neravenstv S. N. Bernshteina”, Dokl. AN SSSR, 60:9 (1948), 1511–1514 | MR | Zbl

[35] A. F. Timan, Theory of approximation of functions of a real variable, Pergamon Press, Oxford–New York, 1963 | MR | MR | Zbl