Aveiro method in reproducing kernel Hilbert spaces under complete dictionary

Abstract

Aveiro method is a sparse representation method in reproducing kernel Hilbert spaces, which gives orthogonal projections in linear combinations of reproducing kernels over uniqueness sets. It, however, suffers from determination of uniqueness sets in the underlying reproducing kernel Hilbert space. In fact, in general spaces, uniqueness sets are not easy to be identified, let alone the convergence speed aspect with Aveiro method. To avoid those difficulties, we propose an new Aveiro method based on a dictionary and the matching pursuit idea. What we do, in fact, are more: The new Aveiro method will be in relation to the recently proposed, the so-called pre-orthogonal greedy algorithm involving completion of a given dictionary. The new method is called Aveiro method under complete dictionary. The complete dictionary consists of all directional derivatives of the underlying reproducing kernels.We show that, under the boundary vanishing condition bring available for the classical Hardy and Paley-Wiener spaces, the complete dictionary enables an efficient expansion of any given element in the Hilbert space. The proposed method reveals new and advanced aspects in both the Aveiro method and the greedy algorithm.

Greedy Algorithms and Rational Approximation in One and Several Variables

“…Based on this work, the authors of [20] propose the so-called "Aveiro Method under complete dictionary (AMUCD)" by combining the ideas of P-OGA with Aveiro Method. Roughly speaking, AMUCD enhances the power of Aveiro Method in that the approximating function given by AMUCD does not require all elements of {p k } ∞ k=1 to be distinct.…”

“…As in AFD, the representation and its derivatives enjoy interpolation properties at {p k } ∞ k=1 . It is shown in [20] that AMUCD is applicable to the classical Hardy spaces and Paley-Wiener spaces. It turns out that AMUCD is, in fact, an alternative representation of AFD.…”

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].…”

Rational Approximation in a Class of Weighted Hardy Spaces

“…AFD thus gives rise to intrinsic positive frequency decomposition of f + and thus that of f as well. Other related work can be seen in [1,20,11]. In the analytic adaptive and positivefrequency approximation aspects the studies of Coifman et al and Qian et al merged together [3,4].…”