Skip to main content
SpringerLink
Log in

Jackson-Type Inequality in Hilbert Spaces and on Homogeneous Manifolds

  • Published:
Analysis Mathematica Aims and scope Submit manuscript

Abstract

We consider a Hilbert space H equipped with a set of strongly continuous bounded semigroups satisfying certain conditions. The conditions allow to define a family of moduli of continuity Ωr(s, f), r ∈ ℕ, s > 0, of vectors in H and a family of Paley—Wiener subspaces PWσ parametrized by bandwidth σ > 0. These subspaces are explored to introduce notion of the best approximation ε(σ, f) of a general vectorin H by Paley—Wiener vectors of a certain bandwidth σ > 0. The main objective of the paper is to prove the so-called Jackson-type estimate ε(σ, f) ≤ C(Ωr(σ−1, f)+σ−r ∥f∥) for σ > 1. Our assumptions are satisfied for a strongly continuous unitary representation of a Lie group G in a Hilbert space H. It allows to obtain the Jackson-type estimates on homogeneous manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. Butzer and H. Berens, Semi-groups of Operators and Approximation, Springer (Berlin, 1967).

    Book  MATH  Google Scholar 

  2. F. Dai, Some equivalence theorems with K-functionals, J. Approx. Theory, 121 (2003) 143–157.

    Article  MathSciNet  MATH  Google Scholar 

  3. F. Dai and Y. Xu, Moduli of smoothness and approximation on the unit sphere and the unit ball, Adv. Math., 224 (2010), 1233–1310.

    Article  MathSciNet  MATH  Google Scholar 

  4. F Dai and Y. Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls, Springer Monographs in Mathematics, Springer (New York, 2013).

    Book  MATH  Google Scholar 

  5. Z. Ditzian, Approximation on Banach spaces of functions on the sphere, J. Approx. Theory, 140 (2006), 31–45.

    Article  MathSciNet  MATH  Google Scholar 

  6. Z. Ditzian, Jackson-type inequality on the sphere, Acta Math. Hungar., 102 (2004), 1–35.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. G. Feichtinger, H. Fuhr and I. Z. Pesenson, Geometric space-frequency analysis on manifolds, J. Fourier Anal. Appl., 22 (2016), 1294–1355.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Helgason, Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Monographs, vol. 39, American Mathematical Society (Providence, RI, 1994).

    MATH  Google Scholar 

  9. S. Krein and I. Pesenson, Interpolation Spaces and Approximation on Lie Groups, The Voronezh State University (Voronezh, 1990).

    MATH  Google Scholar 

  10. S. Krein, Y. Petunin and E. Semenov, Interpolation of Linear Operators, Translations of Mathematical Monographs, vol. 54, Amer. Math. Soc. (Providence, RI, 1982).

    Google Scholar 

  11. V. Kumar and M. Ruzhansky, A note on K-functional, modulus of smoothness, Jackson theorem and Nikolskii-Stechkin inequality on Damek-Ricci spaces, arXiv:2020 (2020).

  12. E. Nelson, Analytic vectors, Ann. of Math., 70 (1959), 572–615.

    Article  MathSciNet  MATH  Google Scholar 

  13. E. Nelson and W. Stinespring, Representation of elliptic operators in an enveloping algebra, Amer. J. Math., 81 (1959), 547–560.

    Article  MathSciNet  MATH  Google Scholar 

  14. S. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems, Springer (Berlin, 1975).

    Book  Google Scholar 

  15. E. Nursultanov, M. Ruzhansky and S. Tikhonov, Nikolskii inequality and Besov, Triebel-Lizorkin, Wiener and Beurling spaces on compact homogeneous manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 16 (2016), 981–1017.

    MathSciNet  MATH  Google Scholar 

  16. E. D. Nursultanov, M. V. Ruzhansky and S. Yu. Tikhonov, Nikolskii inequality and functional classes on compact Lie groups, Funktsional. Anal. i Prilozhen., 49 (2015), 83–87 (in Russian); translation in Funct. Anal. Appl., 49 (2015), 226–229.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. El Ouadih, An equivalence theorem for a K-functional constructed by Beltrami—Laplace operator on symmetric spaces, J. Pseudo-Differ. Oper. Appl., 11 (2020), 1951–1962.

    Article  MathSciNet  MATH  Google Scholar 

  18. I. Pesenson, Interpolation spaces on Lie groups, Dokl. Akad. Nauk SSSR, 246 (1979), 1298–1303 (in Russian).

    MathSciNet  Google Scholar 

  19. I. Pesenson, Nikolski-Besov spaces connected with representations of Lie groups, Dokl. Akad. Nauk SSSR, 273 (1983), 45–49 (in Russian).

    MathSciNet  Google Scholar 

  20. I. Pesenson, The best approximation in a representation space of a Lie group, Dokl. Akad. Nauk USSR, 302 (1988), 1055–1059 (in Russian); translation in Soviet Math. Dokl., 38 (1989), 384–388.

    Google Scholar 

  21. I. Pesenson, On the abstract theory of Nikolski—Besov spaces, Izv. Vyssh. Uchebn. Zaved. Mat., 1988, 59–68 (in Russian); translation in Soviet Math. (Iz. VUZ), 32 (1988), 80–92.

  22. I. Pesenson, Approximations in the representation space of a Lie group, Izv. Vyssh. Uchebn. Zaved. Mat., 1990, 43–50 (in Russian); translation in Soviet Math. (Iz. VUZ), 34 (1990), 49–57.

    Google Scholar 

  23. I. Pesenson, Lagrangian Splines, Spectral Entire Functions and Shannon-Whittaker Theorem on Manifolds, Temple University Research Report 95–87 (1995), 1–28.

  24. I. Pesenson, Sampling of Paley-Wiener functions on stratified groups, J. Fourier Anal. Appl., 4 (1998), 269–280.

    Article  MathSciNet  MATH  Google Scholar 

  25. I. Pesenson, Sobolev, Besov and Paley-Wiener vectors in Banach and Hilbert spaces, Functional Analysis and Geometry: Selim Grigorievich Krein Centennial, Contemp. Math., vol. 733, Amer. Math. Soc. (Providence, RI, 2019), pp. 251–272.

    Chapter  Google Scholar 

  26. N. Ja, Vilenkin, N.J., Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs vol. 22, Amer. Math. Soc. (Providence, RI, 1978).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, PA, 19122, USA

    I. Z. Pesenson

Authors
  1. I. Z. Pesenson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. Z. Pesenson.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pesenson, I.Z. Jackson-Type Inequality in Hilbert Spaces and on Homogeneous Manifolds. Anal Math 48, 1153–1168 (2022). https://doi.org/10.1007/s10476-022-0176-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10476-022-0176-0

Key words and phrases

Mathematics Subject Classification

Search

Navigation