Shift-invariant Spaces Research Articles

Algebraic aspects of two-dimensional convolutional codes

Two-dimensional (2D) codes are introduced as linear shift-invariant spaces of admissible signals on the discrete plane. Convolutional and, in particular, basic codes are characterized both in terms of their internal properties and by means of their input-output representations. The algebraic structure of the class of all encoders that correspond to a given convolutional code is investigated and the possibility of obtaining 2D decoders, free from catastrophic errors, as,veil as efficient syndrome decoders is considered. Some aspects of the state space implementation of 2D encoders and decoders via (finite memory) 2D system are discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Multiresolution and wavelets

Multiresolution is investigated on the basis of shift-invariant spaces. Given a finitely generated shift-invariant subspaceSofL2(ℝd), letSkbe the 2k-dilate ofS(k∈ℤ). A necessary and sufficient condition is given for the sequence {Sk}k∈ℤto fom a multiresolution ofL2(ℝd). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skew-symmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.

Approximation from Shift-Invariant Subspaces of L 2 (ℝ d )

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.

The Structure of Finitely Generated Shift-Invariant Spaces in L2( [formula omitted]d)

A simple characterization is given of finitely generated subspaces of L 2( R d ) 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 from shift-invariant subspaces of 𝐿₂(𝐑^{𝐝})

A complete characterization is given of closed shift-invariant subspaces of L 2 ( R d ) {L_2}({\mathbb {R}^d}) 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.

A Bernstein-type inequality associated with wavelet decomposition

Wavelet decomposition and its related nonlinear approximation problem are investigated on the basis of shift-invariant spaces of functions. In particular, a Bernstein-type inequality associated with wavelet decomposition is established in such a general setting. Several examples of piecewise polynomial spaces are given to illustrate the general theory.

On the construction of multivariate (pre)wavelets

A new approach for the construction of wavelets and prewavelets on R d from multiresolution is presented. The method uses only properties of shift- invariant spaces and orthogonal projectors from L2(R d) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multi- resolution. We present a new approach for the construction of wavelets and prewavelets on R a from multiresolution. Our method, which is based on our earlier work (BDR), (BDR1), uses only properties of shift-invariant spaces and orthogonal projectors from L2(R a) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows us to derive in a simple way previous constructions of wavelets, as well as new constructions, and to settle completely certain basic questions about multiresolution. A univariate function ~ E L2(R ) is called an orthogonal wavelet if its normalized, translated dilates ~kj.k:= 2k/2r j, k rZ, form an orthonormal basis for L~(R). In other words, this system is complete and satisfies the orthogonality conditions

Multilevel preconditioning

This paper is concerned with multilevel techniques for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems. A general estimate is derived which is based on the characterization of Besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions. The result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers. In particular, our analysis indicates how to realize even uniformly bounded condition numbers. For example, the general results are used to show that the Bramble-Pasciak-Xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform. Furthermore, they are applied to a general multivariate setting of refinable shift-invariant spaces, in particular, covering those induced by various types of wavelets.

Fourier analysis of the approximation power of principal shift-invariant spaces

The approximation order provided by a directed set {S h } h>0 of spaces, each spanned by thehZ d -translates of one function, is analyzed. The “nearoptimal” approximants of [R2] from eachs 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 {s h } is obtained froms 1 by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions.