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Approximations of Functions on Manifolds

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Function Spaces, Differential Operators and Nonlinear Analysis

Part of the book series: Teubner-Texte zur Mathematik ((TTZM,volume 133))

Abstract

The article presents a review of investigations by the authors and some other Soviet mathematicians with respect to the approximation of functions on a sphere and on more general manifolds. Of special interest in this context are functions with differential properties, say functions belonging to H rp and B rp,q .

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References

  1. Nikolskij, S. M. Approximation of functions of several variables and imbedding theorems. ( Russian) Nauka, Moscow, 1969

    Google Scholar 

  2. Nikolskij, S. M. A theorem concerning approximation on the sphere. Analysis Mathematica, 9, 1983, 207–221

    Article  MathSciNet  Google Scholar 

  3. Nikolskij, S. M., Lizorkin, P. I., Approximation with spherical polynomials (Russian) Trudy Mat. Inst. Steklov., 166, 1984, 186–200

    Google Scholar 

  4. Nikolskij, S. M., Lizorkin, P. I., Inequalities for harmonical, spherical and algebraical polynomials (Russian) Dokl. Akad. Nauk SSSR, 289 (3), 1986, 541–545

    MathSciNet  Google Scholar 

  5. Nikolskij, S. M., Lizorkin, P. I., Function spaces on a sphere and the theory of approximations. (Russian) Mat. Zam., 41 (4), 1987, 509–516

    Google Scholar 

  6. Nikolskij, S. M., Lizorkin, P. I., A new way to the theory of funftion spaces on the sphere. (Russian) Trudy Mat. Inst. Steklov., 181, 1988, 213–221

    Google Scholar 

  7. Nikolskij, S. M., Lizorkin, P. I., Approximation on a sphere–a survey. Banach center publications, Warsaw, 22, 1989, 281–292

    Google Scholar 

  8. Terechin, A. P., Uniform approximation with algebraic polynomials on a sphere of a noncountably dimensional space. (Russian) Mat. Zam., 41 (3), 1987, 333–341

    Google Scholar 

  9. Terechin, A. P., Approximations with spherical harmonics and classes of differentiable functions on a sphere. (Russian)

    Google Scholar 

  10. Butzer, P., Johnen, H., Lipschitz spaces on compact manifolds. J. Funct. Anal., 7, 1971, 242–266

    Article  MathSciNet  MATH  Google Scholar 

  11. Kaljabin, G. A., On moduli of smoothness of functions on a sphere. (Russian) Dokl. Akad. Nauk SSSR, 294 (5), 1987, 1051–1054

    MathSciNet  Google Scholar 

  12. Rustamov, H. P., On approximation of functions on a sphere. (Russian) Dokl. Akad. Nauk SSSR, 320 (6), 1991, 1319–1325

    Google Scholar 

  13. Lizorkin, P. I., Rustamov, H. P., Nikol’skij-Besov spaces on a sphere and the theory of approximation. (Russian) Trudy Mat. Inst. Steklov., 204, 1992

    Google Scholar 

  14. Terechin, A. P., The bounded group of operators and best possible approximation. (Russian) Diff. uravnenija i vyceslitelnaja mat., Saratov. Univ., 2, 1975, 3–28

    Google Scholar 

  15. Rustamov, H. P., On the equivalency of several moduli of smothness on a sphere. (Russian) Dokl. Akad. Nauk SSSR, 321 (1), 1991, 23–27

    Google Scholar 

  16. Sanjkov, V. Ju., Coercitive properties of degenerate equations. (Russian) Dokl. Akad. Nauk SSSR, 281 (3), 1985, 1320–1322

    Google Scholar 

  17. Nikolskij, S. M., Approximation of functions with trigonometric polynomials on manifolds. (Russian) Dokl. Akad. Nauk SSSR, part 1: 319(6), 1991, part 2: 320 (1), 1991, 40–44

    Google Scholar 

  18. Nikol’skij, S. M., Approximation of functions from the Besov class on manifolds with entire functions of exponential type. (Russian) Dokl. Akad. Nauk SSSR, 319 (1), 1991

    Google Scholar 

  19. Nikol’skij, S. M., Approximation with entire functions of exponential type and with polynomials. (Russian) Trudy Mat. Inst. Steklov., 201, 1992

    Google Scholar 

  20. Nikol’skij, S. M., Again about the approximation with entire functions of exponential type and with polynomials. (Russian) Trudy Mat. Inst. Steklov., 204, 1993 (to appear)

    Google Scholar 

  21. Nikol’skij, S. M., Approximation with entire functions of exponential type in a Banach space metric. (Russian) Dokl. Akad. Nauk Rossii, 324 (2), 1992

    Google Scholar 

  22. Nikolskij, S. M., On a approximation method for functions with polynomials on a manifold. (Russian) Dokl. Akad. Nauk Rossii, 324 (3), 1992, 529–532

    Google Scholar 

  23. Nikolskij, S. M., Properties of several classes of functions on differentiable manifolds. (Russian) Mat. Sb., 33(75), no. 2, 1953, 261–325

    Google Scholar 

  24. Helgason, S., Differential geometry, Lie groups and symmetric spaces. Academic Press, New York, 1978

    MATH  Google Scholar 

  25. Ivanov, V. A., Approximation of functions on homogeneous spaces. ( Russian) PhD thesis, Moskov. Gos. Univ., 1984

    Google Scholar 

  26. Petrova, I. V., Approximation on the hyperboloid in the L2-metric. (Russian) Trudy Mat. Inst. Steklov., 194, 1992, 215–228

    Google Scholar 

  27. Lizorkin, P. I., Direct and inverse theorems in the theory of approximation of functions on the Lobacevskii space. (Russian) Trudy Mat. Inst. Steklov., 194, 1992, 120–147

    Google Scholar 

  28. Vilenkin, H. Ja., Special functions and representation theory of groups. ( Russian) Nauka, Moscow, 1965

    Google Scholar 

  29. Lizorkin, P. I., Approximation on Riemannian manifolds. (Russian) Trudy Mat. Inst. Steklov., 200, 1991, 222–235

    Google Scholar 

  30. Lizorkin, P. I., Guliev, V. S., Function spaces and problems of approximation on the Heisenberg group. (Russian) Trudy Mat. Inst. Steklov., 201, 1992, 245–272

    Google Scholar 

  31. Kamzolov, A. I., On the best approximation in Wp (S’) with polynomials and spherical harmonics. (Russian) Mat. Zam., 32 (3), 1982, 285–293

    MathSciNet  MATH  Google Scholar 

  32. Iiziasvili, L. V., Topurija, S. B., Fourier-Laplace series on the sphere. (Russian) Itogi Nauki i Techn., Ser. Mat. Anal., 15, 1977, 83–130

    Google Scholar 

  33. Kusnirenko, G. G., Problems of approximation of continuous functions on the sphere with finite spherical sums. (Russian) Trudy Charkov. Polytechn. Inst., Ser. Ing. -Phys., 25 (3), 1959, 3–22

    Google Scholar 

  34. Dzlafarov, Ar. S., On spherical analoga of the classical theorems of J. Jackson and S. N. Bernstein. (Russian) Dokl. Akad. Nauk SSSR, 203 (2), 1972, 278–281

    MathSciNet  Google Scholar 

  35. Berens, H., Butzer, P. L., Pawelke, S., Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen. Publ. RIMS, Kyoto Univ., Ser. A, 4 (2), 1968, 201–268

    Article  MathSciNet  MATH  Google Scholar 

  36. Pawelke, S., Über die Approximationsordnung bei Kugelfunktionen und algebraischen Polynomen. Tôhoku Math. J., 24, 1972, 475–486

    MathSciNet  Google Scholar 

  37. Li Luoqing, Approximation for spherical functions by Bochner-Riesz means. Approx. Theory and its Appl., 7 (4), 1991, 93–120

    MATH  Google Scholar 

  38. Bondioli, Cr., Function spaces of Nikol’skij type on compact manifolds. Rendiconti Lincei, Mat. e appl. ser. IX., 14 (3), 1992, 185–194

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Steklov Mathematical Institute RAN, Vavilov 42, 117 966, Moscow GSP-1, Russia

    Pjotr I. Lizorkin & Sergej M. Nikol’skij

Authors
  1. Pjotr I. Lizorkin
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  2. Sergej M. Nikol’skij
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Editor information

Editors and Affiliations

  1. Friedrich-Schiller-University, Jena, Germany

    Hans-Jürgen Schmeisser  & Hans Triebel  & 

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© 1993 Springer Fachmedien Wiesbaden

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Lizorkin, P.I., Nikol’skij, S.M. (1993). Approximations of Functions on Manifolds. In: Schmeisser, HJ., Triebel, H. (eds) Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte zur Mathematik, vol 133. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-11336-2_5

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  • DOI: https://doi.org/10.1007/978-3-663-11336-2_5

  • Publisher Name: Vieweg+Teubner Verlag, Wiesbaden

  • Print ISBN: 978-3-8154-2045-4

  • Online ISBN: 978-3-663-11336-2

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