Approximation Orders of FSI Spaces in L 2 (R d ) (original) (raw)
Related papers
Approximation from shift-invariant subspaces of Lsb2(boldRspd)L\sb 2(\bold R\sp d)Lsb2(boldRspd)
Transactions of the American Mathematical Society, 1994
A complete characterization is given of closed shift-invariant subspaces of L2(Rd) which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace. of all finite linear combinations of shifts of a single function </>. We call its L2(Rd)-closure the principal shift-invariant space generated by cf> and denote it by y ((/,). Of course, a closed shift-invariant subspace of L2(R) need not be principal; it need not even be generated by the shifts of finitely many functions. Shift-invariant spaces are important in a number of areas of analysis. Many spaces, encountered in approximation theory and in finite element analysis,
A survey on L2-approximation orders from shift-invariant spaces
Multivariate Approximation and Applications, 2001
This paper aims at providing a self-contained introduction to notions and results connected with the L 2-approximation power of nitely generated shift-invariant spaces (FSI spaces) S L 2 (R d). Here, approximation order refers to a scaling parameter and to the usual scaling of the L 2-projector onto S , where = f 1 ; : : : ; n g L 2 (R d) is a given set of functions, the so-called generators of S. Special attention is given to the PSI case where the shift-invariant space is generated from the multiinteger translates of just one generator; this case is interesting enough due to its possible applications in wavelet methods. The general FSI case is considered subject to a stability condition being satis ed, and the recent results on so-called superfunctions are developed. For the case of a renable system of generators the sum rules for the matrix mask and the zero condition for the mask symbol, as well as invariance properties of the associated subdivision and transfer operator are discussed. References to the literature and further notes are extensively given at the end of each section. In addition to this, the list of references is enlarged in order to give a rather comprehensive overview on existing literature in the eld.
The Structure of Finitely Generated Shift-Invariant Spaces in
1992
A simple characterization is given of finitely generated subspaces of L2(Rd) which are invariant under translation by any (multi)integer, and is used to give conditions under which such a space has a particularly nice generating set, namely a basis, and, more than that, a basis with desirable properties, such as stability, orthogonality, or linear independence. The last property makes sense only for "local" spaces, i.e., shift-invariant spaces generated by finitely many compactly supported functions, and special attention is paid to such spaces. As an application, we prove that the approximation order provided by a given local space is already provided by the shift-invariant space generated by just one function, with this function constructible as a finite linear combination of the finite generating set for the whole space, hence compactly supported. This settles a question of some 20 years′ standing.
Approximation orders of shift-invariant subspaces of
Journal of Approximation Theory, 2005
We extend the existing theory of approximation orders provided by shift-invariant subspaces of L 2 to the setting of Sobolev spaces, provide treatment of L 2 cases that have not been covered before, and apply our results to determine approximation order of solutions to a refinement equation with a higher-dimensional solution space.
Fourier analysis of the approximation power of principal shift-invariant spaces
Constructive Approximation, 1992
The approximation order provided by a directed set fs h g h>0 of spaces, each spanned by the hZZ d-translates of one function, is analyzed. The \near-optimal" approximants of R2] from each s h to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical Strang-Fix conditions are extended to bounded summable generating functions. The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale fs h g is obtained from s 1 by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive de nite generating functions.
Approximation by group invariant subspaces
Journal de Mathématiques Pures et Appliquées, 2020
In this article we study the structure of Γ-invariant spaces of L 2 (R). Here R is a second countable LCA group. The invariance is with respect to the action of Γ, a non commutative group in the form of a semidirect product of a discrete cocompact subgroup of R and a group of automorphisms. This class includes in particular most of the crystallographic groups. We obtain a complete characterization of Γ-invariant subspaces in terms of range functions associated to shift-invariant spaces. We also define a new notion of range function adapted to the Γ-invariance and construct Parseval frames of orbits of some elements in the subspace, under the group action. These results are then applied to prove the existence and construction of a Γ-invariant subspace that best approximates a set of functional data in L 2 (R). This is very relevant in applications since in the euclidean case, Γ-invariant subspaces are invariant under rigid movements, a very sought feature in models for signal processing.
Optimal Shift Invariant Spaces and Their Parseval Generators
2006
Given a set of functions F={f_1,...,f_m} of L2(Rd), we study the problem of finding the shift-invariant space V with n generators {phi_1,...,phi_n} that is ``closest'' to the functions of F in the sense that V minimize the least square distance from the data F to V over the set of all shift-invariant spaces that can be generated by n or
An approximation problem in multiplicatively invariant spaces
Functional Analysis, Harmonic Analysis, and Image Processing, 2017
Let H be Hilbert space and (Ω, m) a σ-finite measure space. Multiplicatively invariant (MI) spaces are closed subspaces of L 2 (Ω, H) that are invariant under point-wise multiplication by functions from a fixed subset of L ∞ (Ω). Given a finite set of data F ⊆ L 2 (Ω, H), in this paper we prove the existence and construct an MI space M that best fits F , in the least squares sense. MI spaces are related to shift-invariant (SI) spaces via a fiberization map, which allows us to solve an approximation problem for SI spaces in the context of locally compact abelian groups. On the other hand, we introduce the notion of decomposable MI spaces (MI spaces that can be decomposed into an orthogonal sum of MI subspaces) and solve the approximation problem for the class of these spaces. Since SI spaces having extra invariance are in one-to-one relation to decomposable MI spaces, we also solve our approximation problem for this class of SI spaces. Finally we prove that translation-invariant spaces are in correspondence with totally decomposable MI spaces.
Journal of the London Mathematical Society, 1992
Given a compactly supported function ϕ : IR s → C and the space S spanned by its integer translates, we study quasiinterpolants which reproduce (entirely or in part) the space H of all exponentials in S. We do this by imitating the action on H of the associated semi-discrete convolution operator ϕ * by a convolution λ * , λ being a compactly supported distribution, and inverting λ * |H by another local convolution operator µ * . This leads to a unified theory for quasiinterpolants on regular grids, showing that each specific construction now in the literature corresponds to a special choice of λ and µ. The natural choice λ = ϕ is singled out, and the interrelation between ϕ * and ϕ * is analyzed in detail.