Sparse representation of approximation to identity

Abstract

The Dirac-[Formula: see text] distribution may be realized through sequences of convolutions, the latter being also regarded as approximation to the identity. This paper proposes the real form of pre-orthogonal adaptive Fourier decomposition (POAFD) method to realize fast approximation to the identity. It belongs to sparse representation of signals having potential applications in signal and image analysis.

Adaptive Fourier decomposition‐type sparse representations versus the Karhunen–Loève expansion for decomposing stochastic processes

“…(i) The coarse structure:  = L 2 (𝜕D) and the elements of the dictionary  are indexed by all q in D. Examples of such model can include  being the collection of the Poisson kernels in the unit disc or the unit ball or those in the upper-half space. In the upper-half space, case  can be collections of the heat kernels or various kinds of dilated and translated convolution kernels [2]. (ii) The fine structure: Certain functions defined on a region D may constitute a reproducing kernel Hilbert space (RKHS), being denoted as H K , or H 2 (D), and called the Hardy space of the context, where K ∶ D × D → C is the reproducing kernel, satisfying K q (p) = K(p, q) for any pair p, q ∈ D. The related theory and examples may be found in previous studies [2][3][4], as well as in Yang [5].…”

AFD Types Sparse Representations vs. the Karhunen-Loeve Expansion for Decomposing Stochastic Processes

“…Alternatively, D can be the collection of the heat kernels in the upper-half space D 2 . The upper-half space can also be associated with general dilated and translated convolution kernels ( [20]).…”

“…The fine structure: Certain functions defined on a region D may constitute a reproducing kernel Hilbert space (RKHS), be denoted by H K , or H 2 (D), and called the Hardy space of the context, where K : D × D → C is the reproducing kernel, K q (p) = K(p, q), p, q ∈ D. In such case the boundary values, or the non-tangential boundary limits, as often occur in harmonic analysis, span a dense class of H = L 2 (∂D) (as a new notation), or sometimes just dense in a proper subspace of H. In the case the boundary value mapping induces an isometry, or a bounded linear operator, between H 2 (D) and H 2 (∂D). The related theory and examples are contained in [27], [16], [20], as well as in [32]. The relevant literature address various types of RKHSs, or Hardy spaces in such setting, including, for instance, the H 2 space of complex holomorphic functions in the unit disc D 1 , and the h 2 space of harmonic functions in the upper-half Euclidean space, precisely,…”

“…With such formulation Core AFD can be extended to various contexts in which a practical Blaschke product theory may not be known or may not exist. Significant generalizations include POAFD for product dictionary ([14]), POAFD for quaternionic space ( [25]), POAFD for multivariate real variables in the Clifford algebra setting ( [30]), POAFD for weighted Bergman and weighted Hardy spaces ( [21,22]), and most recently sparse representations for the Dirac δ function [20], and etc. We note that the maximal selection principle of POAFD guarantees that, in the optimal one-step selection category, the greediest algorithm, and, in particular, more greedy than what is called orthogonal greedy algorithm ([29]) due to the relation:…”

“…SPOAFD, as an extension of SAFD, can be associated with any BVC dictionary. As for the deterministic case studied in [20], such flexibility makes SPOAFD to be a convenient tool to solve Dirichlet boundary value and Cauchy initial value problems with random data ( [32]): We take the dictionaries of the shifted and dilated Poisson and heat kernels, respectively, and make use the lifting up technology based on the semigroup properties of the kernels, to get the solutions of the random boundary and initial value problems. The same method can be used to solve other types deterministic and random boundary and initial value problems ( [4]).…”