Skip to main content
Log in

Adaptive Fourier series—a variation of greedy algorithm

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


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.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others


  1. Akcay, H., Niness, B.: Orthonormal basis functions for modelling continuous-time systems. Signal Process. 77, 261–274 (1999)

    Article  MATH  Google Scholar 

  2. Bultheel, A., Carrette, P.: Takenaka-Malmquist basis and general Toeplitz matrices. SIAM J. Opt. 41, 1413–1439 (2003)

    MathSciNet  MATH  Google Scholar 

  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)

  4. Davis, G., Mallat, S., Avellaneda, M.: Adaptive greedy approximations. Constr. Approx. 13, 57–98 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Garnett, J.B.: Bounded Analytic Functions. Academic, New York (1987)

    Google Scholar 

  6. Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball. GTM, Springer, New York (2005)3

    Google Scholar 

  7. Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. GTM, Springer, New York (2000)

    MATH  Google Scholar 

  8. Mallat, S., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41, 3397–3415 (1993)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Partington, J.R.: Interpolation, Identification and Sampling, pp. 44–47. Clarendon Press, Oxford (1997)

    MATH  Google Scholar 

  11. Heuberger, P.S.C., Van den Hof, P.M.J., Wahlberg, B.: Modelling and Identification with Rational Orthogonal Basis Functions. Springer, London (2005)

    Book  Google Scholar 

  12. Qian, T.: Mono-components for decomposition of signals. Math. Methods Appl. Sci. 29, 1187–1198 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Qian, T.: Boundary derivative of the phases of inner and outer functions and applications. Math. Methods Appl. Sci. 32, 253–263 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  15. Qian, T., Wang, R., Xu, Y.S., Zhang, H.Z.: Orthonormal bases with nonlinear phases. Adv. Comput. Math. 33, 75–95 (2010)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Temlyakov, V.N.: Greedy algorithm and m-term trigonometric approximation. Constr. Approx. 107, 569–587 (1998)

    Article  MathSciNet  Google Scholar 

  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. Walsh, J.L.: Interpolation and Approximation by Rational Functions in the Complex Plane. AMS (1969)

  20. Wang, R., Xu, Y.S., Zhang, H.Z.: Fast non-linear Fourier expansions. AADA 1(3), 373–405 (2009)

    MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Tao Qian.

Additional information

Communicated by Yuesheng Xu.

The work was supported by Macao FDCT 014/2008/A1 and research grant of the University of Macau No. RG-UL/07-08s/Y1/QT/FSTR.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Qian, T., Wang, YB. Adaptive Fourier series—a variation of greedy algorithm. Adv Comput Math 34, 279–293 (2011).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classifications (2010)