Reproducing kernel approximation in weighted Bergman spaces: Algorithm and applications

Abstract

In this paper, we present algorithms of preorthogonal adaptive Fourier decomposition (POAFD) in weighted Bergman spaces. POAFD, as has been studied, gives rise to sparse approximations as linear combinations of the corresponding reproducing kernels. It is found that POAFD is unavailable in some weighted Hardy spaces that do not enjoy the boundary vanishing condition; as a result, the maximal selections of the parameters are not guaranteed. We overcome this difficulty with two strategies. One is to utilize the shift operator while the other is to perform weak POAFD. In the cases when the reproducing kernels are rational functions, POAFD provides rational approximations. This approximation method may be used to 1D signal processing. It is, in particular, effective to some Hardy Hp space functions for p not being equal to 2. Weighted Bergman spaces approximation may be used in system identification of discrete time‐varying systems. The promising effectiveness of the POAFD method in weighted Bergman spaces is confirmed by a set of experiments. A sequence of functions as models of the weighted Hardy spaces, being a wider class than the weighted Bergman spaces, are given, of which some are used to illustrate the algorithm and to evaluate its effectiveness over other Fourier type methods.

n-Best Kernel Approximation in Reproducing Kernel Hilbert Spaces

“…K a 1 attains its global maximum over all possible choices of the parameter in D. This is what we called Maximal Selection Principle (MSP) in the previous related studies ( [45,49,50]). The proof is divided into two steps:…”

“…The n-best existence results for the Hardy-Sobolev spaces for 0 < β ≤ 1 (β = 1 corresponds to the Dirichlet space) are obtained as a consequence of the main result of this paper (see §4). The Hardy-Sobolev spaces for β > 1 do not fall into the category of the RKHSs considered in the main theorem of this paper, but we show that they are governed by the Sobolev Embedding Theorem (also see [50]).…”

“…Recent studies on approximation in Hardy spaces, including the latest n-best and stochastic AFD approximations, may be found in a sequence of articles [45,43,1,2,15,16,47,62,44,51]. Some early studies of adaptive Fourier decomposition in Bergman and weighted Bergman spaces are given in [49,50]. Celebrating results for the n-best type approximation in weighted Bergman spaces are presented in [52].…”

Sparse representations in reproducing kernels of operator equations

“…POAFD 算法的具体的例子可参见文献 [7,13,14,23]. AFD 和 POAFD 算法在信号与图像处理及 在系统辨识或模型约化 (model reduction) 实际问题中的应用可参见文献 [32][33][34][35][36].…”

Sparse Representations of Solutions to a class of Random Boundary Value Problems

“…The method can be extended to a general class of Hilbert spaces, to form what we call pre-orthogonal stochastic adaptive Fourier decompositions (POSAFDs) ( [42]). The SAFD methods to be emphasised in this paper belong to what is called adaptive Fourier decomposition (AFD), the latter has been undergoing sophisticated developments with a number of variations including adaptive Fourier decomposition (AFD, see [52,41]), preorthogonal adaptive Fourier decomposition (POAFD, suitable for contexts in which a Takenaka-Malmquist (TM) system is unavailable, see [51,43,46,47]), unwinding Blaschke expansion (unwinding AFD, independently developed also by Coifman et al), see [16,14,40,49], n-best reproducing kernel approximation (n-best AFD, see [50,48]), and also in higher dimensions for matrix-valued functions ( [2,3]). The study of the AFD type algorithms was originated from signal decomposition into meaningful positive instantaneous frequency in physics ( [52]).…”