Advances in Computational Mathematics

, Volume 34, Issue 3, pp 279–293 | Cite as

Adaptive Fourier series—a variation of greedy algorithm

  • Tao Qian
  • Yan-Bo Wang


We study decomposition of functions in the Hardy space \(H^2(\mathbb{D} )\) into linear combinations of the basic functions (modified Blaschke products) in the system
$$\label{Walsh like} {B}_n(z)= \frac{\sqrt{1-|a_n|^2}}{1-\overline{a}_{n}z}\prod\limits_{k=1}^{n-1}\frac{z-a_k}{1-\overline{a}_{k}z}, \quad n=1,2,..., $$
where the points a n ’s in the unit disc \(\mathbb{D}\) are adaptively chosen in relation to the function to be decomposed. The chosen points a n ’s do not necessarily satisfy the usually assumed hyperbolic non-separability condition
$$\label{condition} \sum\limits_{k=1}^\infty (1-|a_k|)=\infty $$
in the traditional studies of the system. Under the proposed procedure functions are decomposed into their intrinsic components of successively increasing non-negative analytic instantaneous frequencies, whilst fast convergence is resumed. The algorithm is considered as a variation and realization of greedy algorithm.


Rational orthonormal system Blaschke product Complex hardy space Analytic signal Instantaneous frequency Mono-components Adaptive decomposition of functions Greedy algorithm 

Mathematics Subject Classifications (2010)

42A50 32A30 32A35 46J15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akcay, H., Niness, B.: Orthonormal basis functions for modelling continuous-time systems. Signal Process. 77, 261–274 (1999)zbMATHCrossRefGoogle Scholar
  2. 2.
    Bultheel, A., Carrette, P.: Takenaka-Malmquist basis and general Toeplitz matrices. SIAM J. Opt. 41, 1413–1439 (2003)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bultheel, A., Gonzalez-Vera, P., Hendriksen, E., Njastad, O.: Orthogonal rational functions. In: Cambridge Monographs on Applied and Computational Mathematics, vol. 5. Cambridge University Press (1999)Google Scholar
  4. 4.
    Davis, G., Mallat, S., Avellaneda, M.: Adaptive greedy approximations. Constr. Approx. 13, 57–98 (1997)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Garnett, J.B.: Bounded Analytic Functions. Academic, New York (1987)Google Scholar
  6. 6.
    Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. GTM, Springer, New York (2005)3Google Scholar
  7. 7.
    Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. GTM, Springer, New York (2000)zbMATHGoogle Scholar
  8. 8.
    Mallat, S., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41, 3397–3415 (1993)zbMATHCrossRefGoogle Scholar
  9. 9.
    Ninness, B., Hjalmarsson, H., Gustafsson, F.: Generalized Fourier and Toeplitz results for rational orthonormal bases. SIAM J. Control Optim. 37(2), 429–460 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Partington, J.R.: Interpolation, Identification and Sampling, pp. 44–47. Clarendon Press, Oxford (1997)zbMATHGoogle Scholar
  11. 11.
    Heuberger, P.S.C., Van den Hof, P.M.J., Wahlberg, B.: Modelling and Identification with Rational Orthogonal Basis Functions. Springer, London (2005)CrossRefGoogle Scholar
  12. 12.
    Qian, T.: Mono-components for decomposition of signals. Math. Methods Appl. Sci. 29, 1187–1198 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Qian, T.: Boundary derivative of the phases of inner and outer functions and applications. Math. Methods Appl. Sci. 32, 253–263 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Qian, T., Ho, I.T., Leong, I.T., Wang, Y.B.: Adaptive decomposition of functions into pieces of non-negative instantaneous frequencies. Accepted by International Journal of Wavelets, Multiresolution and Information Processing, vol. 8(5) (2010)Google Scholar
  15. 15.
    Qian, T., Wang, R., Xu, Y.S., Zhang, H.Z.: Orthonormal bases with nonlinear phases. Adv. Comput. Math. 33, 75–95 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Qian, T., Xu, Y.S., Yan, D.Y., Yan, L.X., Yu, B.: Fourier spectrum charaterization of hardy spaces and applications. Proc. Am. Math. Soc. 137, 971–980 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Temlyakov, V.N.: Greedy algorithm and m-term trigonometric approximation. Constr. Approx. 107, 569–587 (1998)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Tan, L.H., Shen, L.X., Yang, L.H.: Rational orthogonal bases satisfying the Bedrosian identity. Adv. Comput. Math. doi: 10.1007/s10444-009-9133-8
  19. 19.
    Walsh, J.L.: Interpolation and Approximation by Rational Functions in the Complex Plane. AMS (1969)Google Scholar
  20. 20.
    Wang, R., Xu, Y.S., Zhang, H.Z.: Fast non-linear Fourier expansions. AADA 1(3), 373–405 (2009)MathSciNetGoogle Scholar
  21. 21.
    Xu, Y.S., Yan, D.Y.: The Bedrosian identity fot the Hilbert transform of product functions. Proc. Am. Math. Soc. 134, 2719–2728 (2006)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and TechnologyUniversity of MacauTaipaChina

Personalised recommendations