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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
Article

Abstract

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
$$\label{Walsh like} {B}_n(z)= \frac{\sqrt{1-|a_n|^2}}{1-\overline{a}_{n}z}\prod\limits_{k=1}^{n-1}\frac{z-a_k}{1-\overline{a}_{k}z}, \quad n=1,2,..., $$
(1)
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
$$\label{condition} \sum\limits_{k=1}^\infty (1-|a_k|)=\infty $$
(2)
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.

Keywords

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 

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Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

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

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