Skip to main content
SpringerLink
Log in

A characterization of Lp(ℝ) by local trigonometric bases with 1<p<∞

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We show that the local trigonometric bases introduced by Malvar, Coifman and Meyer constitute bases, but not unconditional bases, for Lp(ℝ) with 1<p<∞, p≠2. In addition, we characterize the functions in Lp(ℝ) for 1<p<∞ in terms of their local trigonometric basis coefficients.

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. Auscher, G. Weiss and M.V. Wickerhauser, Local sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets, in: Wavelets – A Tutorial in Theory and Applications, ed. C.K. Chui (Academic Press, Boston, 1992) pp. 237–256.

    Google Scholar 

  2. P. Auscher, Remarks on the local Fourier bases, in: Wavelets: Mathematics and Applications, eds. J.J. Benedetto and M.W. Frazier (CRC Press, Boca Raton, FL, 1994) pp. 203–218.

    Google Scholar 

  3. A. Averbuch, E. Braverman and R. Coifman, Efficient computation of oscillatory integrals via adaptive multiscale local Fourier bases, Appl. Comput. Harmon. Anal. 9 (2000) 19–53.

    Article  MathSciNet  Google Scholar 

  4. K. Bittner, Error estimates and reproduction of polynomials for bi-orthogonal local trigonometric bases, Appl. Comput. Harmon. Anal. 6 (1999) 75–102.

    Article  MATH  MathSciNet  Google Scholar 

  5. K. Bittner, Verallgemeinerte Klappoperatoren und Bi-orthogonale Wilson-Basen, Ph.D. thesis, Technische Uniersität München (Shaker Verlag, Aachen, 2000).

  6. L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966) 135–157.

    Article  MATH  MathSciNet  Google Scholar 

  7. C.K. Chui and X. Shi, Characterization of bi-orthogonal cosine wavelets, J. Fourier Anal. Appl. 3 (1997) 559–575.

    MathSciNet  Google Scholar 

  8. R.R. Coifman and Y. Meyer, Remarques sur l'analyse de Fourier à fenêtre, C. R. Acad. Sci. Paris Sér. I 312 (1991) 259–261.

    MathSciNet  Google Scholar 

  9. H.G. Feichtinger, K. Gröchenig and D. Walnut, Wilson bases and modulation space, Math. Nachr. 155 (1992) 7–17.

    MathSciNet  Google Scholar 

  10. L. Grafakos and C. Lennard, Characterization of Lp(ℝn) using Gabor frames, J. Fourier Anal. Appl. 7(2) (2001) 101–126.

    MathSciNet  Google Scholar 

  11. K. Gröchenig and S. Samarah, Non-linear approximation with local Fourier bases, Constr. Approx. 16 (2000) 317–331.

    Article  MathSciNet  Google Scholar 

  12. E. Hernándaz and G.L. Weiss, A First Course on Wavelets (CRC Press, New York, 1996).

    Google Scholar 

  13. R.A. Hunt, On the convergence of Fourier series, in: Proc. of Conf. on Orthogonal Expansions and Their Continuous Analogues, Edwardsville, IL (1967) (Sounthern Illinois Univ. Press, Carbondale, IL, 1968) pp. 235–255.

    Google Scholar 

  14. B. Jawerth, Y. Liu and W. Sweldens, Signal compression with smooth local trigonometric bases, Opt. Engrg. 33 (1994) 2125–2135.

    Article  Google Scholar 

  15. B. Jawerth and W. Sweldens, Bi-orthogonal smooth local trigonometric bases, J. Fourier Anal. Appl. 2 (1995) 109–133.

    Article  MathSciNet  Google Scholar 

  16. R. Larsen, An Introduction to the Theory of Multipliers, Grandlehren der mathematishen Wissenschaften, Vol. 175 (Springer, New York, 1971).

    Google Scholar 

  17. Q. Lian, Y. Wang and D. Yan, Efficient computations of oscillatory singular integrals with local Fourier bases and their error estimates, Preprint.

  18. Q. Lian, Y. Wang and D. Yan, Optimal local trigonometric bases with nonuniform partitions, Acta Math. Sinica, English Series, accepted.

  19. S. Lu, M.H. Taibleson and G. Weiss, On the almost everywhere convergence of Bochner–Riesz means of multiple Fourier series, in: Lecture Notes in Mathematics, Vol. 908 (Springer, New York, 1982) pp. 311–318.

    Google Scholar 

  20. S. Lu and Z. Yan, Criterion on Lp-boundedness for a class of oscillatory singular integral with rough kernel, Rev. Mat. Iberoamericana 2(8) (1992) 201–219.

    Google Scholar 

  21. H.S. Malvar, Lapped transforms for efficient transform/subband coding, IEEE Trans. Acoustic Speech Signal Process. 38 (1990) 969–978.

    Article  Google Scholar 

  22. H.S. Malvar, Signal Processing with Lapped Transforms (Artech House, Norwood, MA, 1992).

    Google Scholar 

  23. G. Matviyenko, Optimized local trigonometric bases, Appl. Comput. Harmon. Anal. 3 (1996) 301–323.

    Article  MATH  MathSciNet  Google Scholar 

  24. E. Wesfreid and M.V. Wickerhauser, Adapted local trigonometric transforms and speech processing, IEEE Trans. Signal Process. 41 (1993) 3596–3600.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, P. R. China

    Qiaofang Lian & Yongge Wang

  2. Department of Applied Mathematics, Beijing University of Aeronautics and Astronautics, Beijing, 100083, P. R. China

    Yongge Wang

  3. School of Information Science and Engineering, the Graduate School of the Chinese Academy of Sciences, Beijing, 100039, P. R. China

    Dunyan Yan

Authors
  1. Qiaofang Lian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongge Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dunyan Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Zhou

Dedicated to Dr. Charles A. Micchelli for his 60th birthday

Mathematics subject classification (2000)

42C15.

Supported by Prof. Y. Xu under his grant in program of “One Hundred Distinguished Chinese Scientists” of the Chinese Academy of Sciences, the National Natural Science Foundation of China (No. 10371122), and the second author is supported by Tianyuan Fund for Mathematics (No. A0324648).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lian, Q., Wang, Y. & Yan, D. A characterization of Lp(ℝ) by local trigonometric bases with 1<p<∞. Adv Comput Math 25, 91–104 (2006). https://doi.org/10.1007/s10444-004-7625-0

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-004-7625-0

Keywords

Search

Navigation