Adaptive Fourier series - a variation of greedy algorithm

We study decomposition of functions in the Hardy space H2(D) into linear combinations of the basic functions (modified Blaschke products) in the system Bn(z) = √ 1 − |an|2 1 − anz n−1 ∏ k=1 z − ak 1 − akz , n = 1, 2, ..., (1) where the points an’s in the unit disc D are adaptively chosen in relation to the function to be decomposed. The chosen points an’s do not necessarily satisfy the usually assumed hyperbolic non-separability condition ∞ ∑ k=1 (1 − |ak|) = ∞ (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. 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. T. Qian (B) · Y.-B. Wang Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macao, China e-mail: [email protected] Y.-B. Wang e-mail: [email protected] 280 T. Qian, Y.-B. Wang