Adaptive Fourier decomposition in

Abstract

Let X be a Banach space. We say D = {gα, α ∈ Γ} ⊂ X is a dictionary of X, if it satisfies ||gα||X = 1, and Span{D} is dense in X. By the terminology adaptive Fourier decomposition (AFD), we refer to approximations in the H2(T) space by using linear combinations of the analytic Szegö kernels. The earlier and main studies on AFD were presented in previous studies1,2 when studying the problem of signal decomposition into mono-components.3,4 In the same spirit but restricted to Hilbert spaces, the adaptive approximation under a dictionary can be found in the study of projection pursuit regression and neural network training.5 In the series studies, S. Mallat gave a matching pursuit algorithm in real Hilbert spaces and applied it to signal decomposition with a so-called time-frequency dictionary.6,7 The Hilbert space matching pursuit algorithm is also called pure greedy algorithm (PGA) being referred to Temlyakov s work.8 Based on PGA, similar algorithms such as weak greedy algorithm (WGA), Chebyshev threshold greedy algorithm (TGA), and relaxed greedy algorithm (RGA) in Hilbert spaces and in real Banach spaces with smoothness were studied.9-12 The present paper gives two AFD-type approximations to functions in the Hp(T) spaces in the unit disc for 1 < p < ∞ other than p = 2. In the sequel, we use D and T for the unit disc and the unit circle, respectively. Below, we give a quick summary of the AFD algorithm for the Hardy space H2(T), where the dictionary consists of the normalized Szegö kernel of the context, namely