Adaptive Fourier series—a variation of greedy algorithm

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.