Optimal shift invariant spaces and their Parseval frame generators

Recovery of missing samples for oversampling in shift invariant spaces

The aim of this paper is to study sampling and reconstruction in a shift invariant space with multiple generators. On contrary to the classical case of a shift invariant space with a single generator, it is shown that cannot be a stable set of sampling for where when Further the problems of perturbation of a stable set of sampling and local reconstruction method are discussed along with an illustration and implementation.

In this paper, we introduce the notion of Walsh shift-invariant space and present a unified approach to the study of shift-invariant systems to be frames in L2(ℝ+). We obtain a necessary condition and three sufficient conditions under which the Walsh shift-invariant systems constitute frames for L2(ℝ+). Furthermore, we discuss applications of our main results to obtain some known conclusions about the Gabor frames and wavelet frames on positive half line.

Erratum to: Periodic operators and elements. Shift operators. Shift invariant spaces

To reconstruct a function from its sampling value is not always exact, error may arise due to a lot of reasons, therefore error estimation is useful in reconstruction. For non-uniform sampling in shift invariant space, three kinds of errors of the reconstruction formula are discussed in this article. For every kind of error, we give an estimation. We find the accuracy of the reconstruction formula mainly depends on the decay property of the generator and the sampling function.

The structure of shift–modulation invariant spaces: The rational case

Extra invariance of shift-invariant spaces on LCA groups

It is the purpose of this paper to feature the link between the theory of minimal norm interpolation over lattices by elements fromSobolev algebras H s (S d )with what is known as the theory of spline-type (or principal shift invariant) spaces. As an extremely useful tool allowing us to establish various kinds of robustness results we will present the Wiener amalgam spacesW(B, l p with general (smooth) local components and globall p behavior. For this reason a summary of their most important properties, including convolution relations and the behavior under the Fourier transform, is presented. The discussion of projection and minimal norm interpolation operators is not restricted to the pure Hilbert space setting for which these concepts were developed originally. Among others we show LP-stability of the (for p = 2 orthogonal) projection from LP onto the corresponding spline-type spaces with lP-coefficients. As a main result (which can be formulated in several different concrete ways) we show that for s > d/2 the mapping f ↦ Qs, a(f), from f to the minimal norm interpolation in Hs over the lattice aℤd, a >, depends continuously on the input parameters (s, a). It also extends to certain fractional LP-Sobolev spaces consisting of continuous functions in LP. In this modified setting the outcome of this procedure depends continuously on (s, a) in the LP sense. Moreover, the mapping is robust against small jitter errors. Wiener amalgam spaces tum out to be very useful, both for a precise formulation and in the proofs of such results.

Backward shift operator and finite dimensional de Branges Rovnyak spaces in the ball

The Kalman filter provides a finite dimensional solution when the signal and observation processes are linear and have Gaussian noise. In this paper the effect of a small non- linearity in the signal is discussed by considering stochastic flows for the signal and a Girsanov transformation for the observation. The result can be expressed in terms of Gaus- sian densities.

The past several years have witnessed a surge of research investigating various aspects of sparse representations and compressed sensing. Most of this work has focused on the finite-dimensional setting in which the goal is to decompose a finite-length vector into a given finite dictionary. Underlying many of these results is the conceptual notion of an uncertainty principle: a signal cannot be sparsely represented in two different bases. Here, we extend these ideas and results to the analog, infinite-dimensional setting by considering signals that lie in a finitely-generated shift-invariant (SI) space. This class of signals is rich enough to include many interesting special cases such as multiband signals and splines. By adapting the notion of coherence defined for finite dictionaries to infinite SI representations, we develop an uncertainty principle similar in spirit to its finite counterpart. We demonstrate tightness of our bound by considering a bandlimited lowpass train that achieves the uncertainty principle. Building upon these results and similar work in the finite setting, we show how to find a sparse decomposition in an overcomplete dictionary by solving a convex optimization problem. The distinguishing feature of our approach is the fact that even though the problem is defined over an infinite domain with infinitely many variables and constraints, under certain conditions on the dictionary spectrum our algorithm can find the sparsest representation by solving a finite-dimensional problem.

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.

This paper presents a method of constructing Parseval frames from any collection of complex envelopes. The resulting Enveloped Sinusoid Parseval (ESP) frames can represent a wide variety of signal types as specified by their physical morphology. Since the ESP frame retains its Parseval property even when generated from a variety of envelopes, it is compatible with large scale and iterative optimization algorithms. The ESP construction provides an analysis and synthesis transform pair that can be tuned according to the characteristics of the signal to which it is applied. This work provides examples of ESP frame generation for both synthetic and experimentally measured signals. Furthermore, the frame's compatibility with distributed sparse optimization frameworks is demonstrated. Numerical experiments on acoustics data reveal that the flexibility of this method allows it to be simultaneously competitive with a signal agnostic short-time Fourier Transform in time-frequency processing and also with Prony's Method for time-constant parameter estimation, surpassing the shortcomings of each individual technique.

We explore connections between boundary representations of operator spaces and those of the associated Paulsen systems. Using the notions of finite representation and separating property which we introduce for operator spaces, the boundary representations for operator spaces are characterized. We also introduce weak boundary for operator spaces. Rectangular hyperrigidity for operator spaces introduced here is used to establish an analogue of Saskin’s theorem in the setting of operator spaces in finite dimensions.

Slice-continuous sets in reflexive Banach spaces: convex constrained optimization and strict convex separation