Skip to main content
SpringerLink
Log in

Parseval Space-Frequency Localized Frames on Sub-Riemannian Compact Homogeneous Manifolds

  • Chapter
  • First Online:
Frames and Other Bases in Abstract and Function Spaces

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

The objective of this chapter is to describe a construction of Parseval bandlimited and localized frames on sub-Riemannian compact homogeneous manifolds.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. T. Coulhon, G. Kerkyacharian, P. Petrushev, Heat kernel generated frames in the setting of Dirichlet spaces. J. Fourier Anal. Appl. 18 (5), 995–1066 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Duffin, A. Schaeffer, A class of nonharmonic Fourier series. Trans. AMS 72, 341–366 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  3. F. Filbir, H. Mhaskar, A quadrature formula for diffusion polynomials corresponding to a generalized heat kernel. J. Fourier Anal. Appl. 16 (5), 629–657 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Geller, I. Pesenson, Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds. J. Geom. Anal. 21 (2), 334–371 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Geller, I. Pesenson, Kolmogorov and linear widths of balls in Sobolev spaces on compact manifolds. Math. Scand. 115 (1), 96–122 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. L. Hörmander, Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Helgason, Differential Geometry and Symmetric Spaces (Academic, New York, 1962)

    MATH  Google Scholar 

  8. S. Helgason, Groups and Geometric Analysis (Academic, New York, 1984)

    MATH  Google Scholar 

  9. P. Maheux, Estimations du noyau de la chaleur sur les espaces homogenes. J. Geom. Anal. 8, 65–96 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Nagel, E. Stein, S. Wainger, Balls and metrics defined by vector fields I: basic properties. Acta Math. 155, 103–147 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  11. I. Pesenson, Interpolation spaces on Lie groups. Dokl. Akad. Nauk SSSR (Russian) 246 (6), 1298–1303 (1979)

    Google Scholar 

  12. I. Pesenson, Nikolskii-Besov spaces connected with representations of Lie groups. Dokl. Akad. Nauk SSSR (Russian) 273 (1), 45–49 (1983); Engl. Transl. in Soviet Math. Dokl. 28 (1983)

    Google Scholar 

  13. I. Pesenson, Abstract theory of Nikolskii-Besov spaces. Izv. Vyssh. Uchebn. Zaved. Mat. 59–70 (1988); Engl. Transl. in Soviet Mathematics, 32 (6) (1988)

    Google Scholar 

  14. I. Pesenson, The best approximation in a representation space of a Lie group. Dokl. Acad. Nauk USSR 302 (5), 1055–1059 (1988); (Engl. Transl. in Soviet Math. Dokl. 38 (2) (1989), 384–388)

    Google Scholar 

  15. I. Pesenson, Approximations in the representation space of a Lie group. Izv. Vyssh. Uchebn. Zaved. Mat. 1990 (7), 43–50 (1990); translation in Soviet Math. (Iz. VUZ) 34 (7), 49–57 (1990)

    Google Scholar 

  16. I. Pesenson, The Bernstein inequality in representations of Lie groups. Dokl. Akad. Nauk SSSR (Russian) 313 (4), 803–806 (1990); translation in Soviet Math. Dokl. 42 (1), 87–90 (1991)

    Google Scholar 

  17. I. Pesenson, Functions that are smooth along vector fields. Mat. Zametki (Russian) 48 (1), 95–102 (1990); translation in Math. Notes 48 (1990), no. 1–2, 683–688 (1991)

    Google Scholar 

  18. I. Pesenson, A sampling theorem on homogeneous manifolds. Trans. Am. Math. Soc. 352 (9), 4257–4269 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. I. Pesenson, Poincare-type inequalities and reconstruction of Paley-Wiener functions on manifolds. J. Geom. Anal. 14 (1), 101–121 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Pesenson, Bernstein-Nikolski inequality and Riesz interpolation formula on compact homogeneous manifolds. J. Approx. Theory 150 (2), 175–198 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. I.Z. Pesenson, D. Geller, Cubature formulas and discrete Fourier transform on compact manifolds, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry. In Memory of Leon Ehrenpreis (Developments in Mathematics, vol. 28), ed. by H.M. Farkas, R.C. Gunning, M.I. Knopp, B.A. Taylor (Springer, New York, 2013)

    Google Scholar 

  22. L. Rothschild, E. Stein, Hypoelliptic differential operators and nilpotent groups. Acta Math. 137, 247–320 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Triebel, Theory of Function Spaces II, Monographs in Mathematics, vol. 84 (Birkhäuser, Basel, 1992)

    Book  Google Scholar 

Download references

Acknowledgements

I am thankful to Hartmut Führ and Gerard Kerkyacharian for stimulating discussions.

Author information

Authors and Affiliations

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

    Isaac Pesenson

Authors
  1. Isaac Pesenson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Isaac Pesenson .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Temple University, Philadelphia, Pennsylvania, USA

    Isaac Pesenson

  2. School of Mathematics and Statistics, The University of New South Wales, Sydney, New South Wales, Australia

    Quoc Thong Le Gia

  3. Department of Mathematics, The Graduate Center, CUNY, New York, New York, USA

    Azita Mayeli

  4. Institute of Mathematical Sciences, Claremont Graduate University, Claremont, California, USA

    Hrushikesh Mhaskar

  5. Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong

    Ding-Xuan Zhou

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Pesenson, I. (2017). Parseval Space-Frequency Localized Frames on Sub-Riemannian Compact Homogeneous Manifolds. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, DX. (eds) Frames and Other Bases in Abstract and Function Spaces. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55550-8_17

Download citation

Publish with us

Policies and ethics

Search

Navigation