Shift-invariant Subspaces Of L2 Research Articles

Shift-invariant spline subspaces of Sobolev spaces and their order of approximation

Closed shift-invariant subspaces of Hilbert space are the core idea of important classes of signal and image analysis methods such as wavelets, shearlets, radial basis functions and many more. In 1994 de Boor, DeVore and Ron [4] gave a full characterization of the order of approximation in such shift-invariant subspaces of L2(Rn), n∈N. Ron and Holtz extended this work in [10] to Sobolev norms, and gave specific criteria on the generating functions to guarantee certain orders of approximation. However, not for all generating functions these criteria are easy to verify. We here give new criteria which are designed for generating functions given explicitly in Fourier domain. We show by various examples on spline functions that in these cases our new criteria give better estimates on the order of approximation.

Approximate oblique dual frames

In representations using frames, oblique duality appears in situations where the analysis and the synthesis has to be done in different subspaces. In some cases, we cannot obtain an explicit expression for the oblique duals and in others there exists only one oblique dual frame which has not the properties we need. Also, in practice the computations are not exact. To give a solution to these problems, in this work we introduce and investigate the notion of approximate oblique dual frames first in the setting of separable Hilbert spaces. We present several properties and provide different characterizations of approximate oblique dual frames. We focus then on approximate oblique dual frames in shift-invariant subspaces of L2(R) and give different conditions on the generators that assure their existence. The importance of approximate oblique dual frames from a numerical and computational point of view is illustrated with an example of frame sequences generated by B-splines, where the previous results are used to construct approximate oblique dual frames which have better attributes than the exact ones. We provide an expression for the approximation error and study its behaviour.

Modeling Sampling in Tensor Products of Unitary Invariant Subspaces

The use of unitary invariant subspaces of a Hilbert spaceHis nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces ofL2(R)and also periodic extensions of finite signals are remarkable examples where this occurs. As a consequence, the availability of an abstract unitary sampling theory becomes a useful tool to handle these problems. In this paper we derive a sampling theory for tensor products of unitary invariant subspaces. This allows merging the cases of finitely/infinitely generated unitary invariant subspaces formerly studied in the mathematical literature; it also allows introducing the several variables case. As the involved samples are identified as frame coefficients in suitable tensor product spaces, the relevant mathematical technique is that of frame theory, involving both finite/infinite dimensional cases.

CONSISTENT SAMPLING WITH MULTI-, PRE- AND POST-FILTERINGS

We consider the approximate consistent sampling process in the space L2(ℝ) of signals of finite energy, where multi-, pre- and post-filters [Formula: see text] and [Formula: see text] are from L2(ℝ). We take {(f * hi)(qk)| 1 ≤ i ≤ M, k ∈ ℤ} as measurements(or generalized samples) of an input signal f and then seek its consistent approximation [Formula: see text] in the reconstruction space V(Φ), which is a shift-invariant subspace of L2(ℝ) generated by N post-filters [Formula: see text]. Here [Formula: see text] is a rational number and the consistency means that f and [Formula: see text] have the same measurements. We find equivalent conditions for the existence of such consistent approximation in L2(ℝ) together with some relevant properties such as performance analysis.

SHIFT GENERATED DUAL FRAMES FOR LOCALLY COMPACT ABELIAN GROUPS

Let G be a metrizable, -compact locally compact abelian group with a compact open subgroup. In this paper we dene the Gramian and the dual Gramian operators for shift invariant subspaces of L2( G) and we use them to characterize shift generated dual frames for shift in- variant spaces, which forms a frame for a subspace of L2( G). We present necessary and sufficient conditions for which standard dual is a unique SG-dual frame of type I and type II.

SHIFT INVARIANT SPACES FOR LOCAL FIELDS

This paper is an investigation of shift invariant subspaces of L2(G), where G is a locally compact abelian group, or in general a local field, with a compact open subgroup. In this paper we state necessary and sufficient conditions for shifts of an element of L2(G) to be an orthonormal system or a Parseval frame. Also we show that each shift invariant subspace of L2(G) is a direct sum of principle shift invariant subspaces of L2(G) generated by Parseval frame generators.

SAMPLING EXPANSION IN SHIFT INVARIANT SPACES

For any ϕ(t) in L2(ℝ), let V(ϕ) be the closed shift invariant subspace of L2(ℝ) spanned by integer translates {ϕ(t - n) : n ∈ ℤ} of ϕ(t). Assuming that ϕ(t) is a frame or a Riesz generator of V(ϕ), we first find conditions under which V(ϕ) becomes a reproducing kernel Hilbert space. We then find necessary and sufficient conditions under which an irregular or a regular shifted sampling expansion formula holds on V(ϕ) and obtain truncation error estimates of the sampling series. We also find a sufficient condition for a function in L2(ℝ) that belongs to a sampling subspace of L2(ℝ). Several illustrating examples are also provided.

Order of linear approximation from shift-invariant spaces

A Fourier analysis approach is taken to investigate the approximation order of scaled versions of certain linear operators into shift-invariant subspaces ofL2(Rd). Quasi-interpolants and cardinal interpolants are special operators of this type, and we give a complete characterization of the order in terms of some type of ellipticity condition for a related function. We apply these results by showing that theL2-approximation order of a closed shift-invariant subspace can often be realized by such an operator.

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.