Shift-invariant Subspaces Research Articles

De Branges-Rovnyak spaces generated by row Schur functions with mate

In this paper, we study the de Branges-Rovnyak spaces H(B) generated by row Schur functions B with mate a. We prove that the polynomials are dense in H(B), and characterize the backward shift invariant subspaces of H(B). We then describe the cyclic vectors in H(B) when B is of finite rank and dim⁡(aH2)⊥<∞.

Shift invariant subspaces in growth spaces and sets of finite entropy

AbstractWe investigate shift invariant subspaces within the realm of analytic functions in the unit disc, whose radial growth is determined by a majorant . Our main result offers a complete characterization of the shift invariant subspaces generated by Nevanlinna class functions within the aforementioned class of growth spaces. Our description is expressed in terms of a certain ‐entropy condition, which naturally arises in connection with boundary zero sets for analytic functions in the unit disc, having modulus of continuity not exceeding on the unit circle. Furthermore, we utilize our findings to establish a structural theorem on approximation in model spaces.

Parameter estimation of hybrid pulse streams based on sub-Nyquist sampling and Shift-invariant subspace projection

Finite Rate of Innovation (FRI) sampling theory offers an efficient approach for the sub-Nyquist measurement of the short pulse stream signal, but it can only address parameter measurement problems under waveform consistency. To overcome the measurement challenge of the hybrid pulse streams, this paper proposes a subspace projection-based FRI parameter estimation method under sub-Nyquist sampling conditions. Our method is based on the concept of subspace decomposition, which involves finely dividing different pulse signals within their respective Shift-invariant (SI) subspaces. We use the generating function of the corresponding space as a decoupling object to achieve parameter estimation through FRI measurement methods. The sampling process is efficient and simple, requiring only single-channel sampling to measure the characteristic parameters of all hybrid pulses. We not only provide theoretical guarantees for the implementation of this approach but also conduct numerous numerical simulations and hardware verification experiments. Our experimental results demonstrate that our method achieves higher parameter measurement accuracy compared to classical methods in a simpler manner.

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 spectral cosynthesis in the harmonically weighted Dirichlet spaces

For a finite positive Borel measure $\mu$ on the unit circle, let $\mathcal{D}(\mu)$ be the associated harmonically weighted Dirichlet space. A shift invariant subspace $\mathcal{M}$ recognizes strong approximate spectral cosynthesis if there exists a sequence of shift invariant subspaces $\mathcal{M}_k$, with finite codimension, such that the orthogonal projections onto $\mathcal{M}_k$ converge in the strong operator topology to the orthogonal projection onto $\mathcal{M}$. If $\mu$ is a finite sum of atoms, then we show that shift invariant subspaces of $\mathcal{D}(\mu)$ admit strong approximate spectral cosynthesis.

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.

Alternative proofs of Mandrekar’s theorem

We present two alternative proofs of Mandrekar’s theorem, which states that an invariant subspace of the Hardy space on the bidisc is of Beurling type precisely when the shifts satisfy a doubly commuting condition [Proc. Amer. Math. Soc. 103 (1988), pp. 145–148]. The first proof uses properties of Toeplitz operators to derive a formula for the reproducing kernel of certain shift invariant subspaces, which can then be used to characterize them. The second proof relies on the reproducing property in order to show that the reproducing kernel at the origin must generate the entire shift invariant subspace.

Average sampling expansions from regular and irregular samples over shift-invariant subspaces on LCA groups

Average sampling expansions from regular and irregular samples over shift-invariant subspaces on LCA groups

Sampling theorem and reconstruction formula for the space of translates on the Heisenberg group

The paper deals with the necessary and sufficient conditions for obtaining reconstruction formulae and sampling theorems for every function belonging to the principal shift invariant subspace of $ L^2(\mathbb{H}^n) $, both in the time domain and a transform domain, where $ \mathbb{H}^n $ denotes the Heisenberg group.

Gabor frames for rational functions

We study the frame properties of the Gabor systems G(g;α,β):={e2πiβmxg(x-αn)}m,n∈Z.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\mathfrak {G}}(g;\\alpha ,\\beta ):=\\{e^{2\\pi i \\beta m x}g(x-\\alpha n)\\}_{m,n\\in {\\mathbb {Z}}}. \\end{aligned}$$\\end{document}In particular, we prove that for Herglotz windows g such systems always form a frame for L^2({mathbb {R}}) if alpha ,beta >0, alpha beta le 1. For general rational windows gin L^2({mathbb {R}}) we prove that {mathfrak {G}}(g;alpha ,beta ) is a frame for L^2({mathbb {R}}) if 0<alpha ,beta , alpha beta <1, alpha beta not in {mathbb {Q}} and {hat{g}}(xi )ne 0, xi >0, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of L^2({mathbb {R}}).

On approximation properties of matrix-valued multi-resolution analyses

We study approximation properties of multi-resolution analyses in the context of matrix-valued function spaces. Here, we generalize the notions of approximation order and density order given by the reference [de Boor C, DeVore RA, Ron A. Approximation from shift-invariant subspaces of . Trans Am Math Soc. 1994;341(2):787–806]. Indeed, we prove a characterization of the closed subspaces generated by the shifts of a single matrix-valued function that provide approximation order and/or density order . To give our conditions, we need the classical notion of approximate continuity. As a consequence, we prove necessary and sufficient conditions on a matrix-valued function to be a scaling function in a multi-resolution analysis.

Single-pixel compressive imaging in shift-invariant spaces via exact wavelet frames

This paper introduces a novel framework for single-pixel imaging via compressive sensing (CS) in shift-invariant (SI) spaces by exploiting the sparsity property of a wavelet representation. We reinterpret the acquisition procedure of a single-pixel camera as filtering of the observed signal with continuous-domain functions that lie in an SI subspace spanned by the integer shifts of the box function. The signal is modeled by an arbitrary SI generator whose special case is the box function, which, as we show in the paper, is conventionally used in single-pixel imaging. We propose to use separable B-spline generators which are intuitively complemented by sparsity-inducing spline wavelets. The SI models of the acquisition and the underlying signal lead to an exact discretization of an inherently continuous-domain inverse problem to a finite-dimensional problem of CS type. By solving the CS optimization problem, a parametric representation of the signal is obtained. Such a representation offers many practical advantages in image processing applications. We propose an efficient matrix-free implementation of the framework and conduct it on the standard test images and real-world measurement data. Experimental results show that the proposed framework achieves a significant improvement of the reconstruction quality relative to the conventional discretization in CS setups. MATLAB implementation of the method described in this paper has been made publicly available on https://github.com/retiro/compressive_imaging_in_si_spaces.

Analogues of finite Blaschke products as inner functions

We give a generalization of the notion of finite Blaschke products from the perspective of generalized inner functions in various reproducing kernel Hilbert spaces. Further, we study precisely how these functions relate to the so-called Shapiro--Shields functions and shift-invariant subspaces generated by polynomials. Applying our results, we show that the only entire inner functions on weighted Hardy spaces over the unit disk are multiples of monomials, extending recent work of Cobos and Seco.

Study of nearly invariant subspaces with finite defect in Hilbert spaces

In this article, we briefly describe nearly $$T^{-1}$$ invariant subspaces with finite defect for a shift operator T having finite multiplicity acting on a separable Hilbert space $${\mathcal {H}}$$ as a generalization of nearly $$T^{-1}$$ invariant subspaces introduced by Liang and Partington in Complex Anal. Oper. Theory 15(1) (2021) 17 pp. In other words, we characterize nearly $$T^{-1}$$ invariant subspaces with finite defect in terms of backward shift invariant subspaces in vector-valued Hardy spaces by using Theorem 3.5 in Int. Equations Oper. Theory 92 (2020) 1–15. Furthermore, we also provide a concrete representation of the nearly $$T_B^{-1}$$ invariant subspaces with finite defect in a scale of Dirichlet-type spaces $${\mathcal {D}}_\alpha $$ for $$\alpha \in [-1,1]$$ corresponding to any finite Blashcke product B, as was done recently by Liang and Partington for defect zero case (see Section 3 of Complex Anal. Oper. Theory 15(1) (2021) 17 pp).

The random convolution sampling stability in multiply generated shift invariant subspace of weighted mixed Lebesgue space

&lt;abstract&gt;&lt;p&gt;In this paper, we mainly investigate the random convolution sampling stability for signals in multiply generated shift invariant subspace of weighted mixed Lebesgue space. Under some restricted conditions for the generators and the convolution function, we conclude that the defined multiply generated shift invariant subspace could be approximated by a finite dimensional subspace. Furthermore, with overwhelming probability, the random convolution sampling stability holds for signals in some subset of the defined multiply generated shift invariant subspace when the sampling size is large enough.&lt;/p&gt;&lt;/abstract&gt;

Shift-Invariant-Subspace Discretization and Volume Reconstruction for Light Field Microscopy

Light Field Microscopy (LFM) is an imaging technique that captures 3D spatial information with a single 2D image. LFM is attractive because of its relatively simple implementation and fast volume acquisition rate. Capturing volume time series at a camera frame rate can enable the study of the behaviour of many biological systems. For instance, it could provide insights into the communication dynamics of living 3D neural networks. However, conventional 3D reconstruction algorithms for LFM typically suffer from high computational cost, low lateral resolution, and reconstruction artifacts. In this work, we study the origin of these issues and propose novel techniques to improve the performance of the reconstruction process. First, we propose a discretization approach that uses shift-invariant subspaces to generalize the typical discretization framework used in LFM. Then, we study the shift-invariant-subspace assumption as a prior for volume reconstruction under ideal conditions. Furthermore, we present a method to reduce the computational time of the forward model by using singular value decomposition (SVD). Finally, we propose to use iterative approaches that incorporate additional priors to perform artifact-free 3D reconstruction from real light field images. We experimentally show that our approach performs better than Richardson-Lucy-based strategies in computational time, image quality, and artifact reduction.

Compressions of kth-order slant Toeplitz operators to model spaces

In this paper, we consider compressions of kth-order slant Toeplitz operators to the backward shift-invariant subspaces of the classical Hardy space H2. In particular, we characterize these operators using compressed shifts and finite-rank operators of special kind.

Average sampling and reconstruction in shift-invariant subspaces of mixed Lebesgue space L^p,q(ℝ^d+1)

Average sampling and reconstruction in shift-invariant subspaces of mixed Lebesgue space L^p,q(ℝ^d+1)

Average sampling in certain subspaces of Hilbert–Schmidt operators on $$L^2(\mathbb {R}^d)$$

The concept of translation of an operator allows to consider the analogue of shift-invariant subspaces in the class of Hilbert–Schmidt operators. Thus, we extend the concept of average sampling to this new setting, and we obtain the corresponding sampling formulas. The key point here is the use of the Weyl transform, a unitary mapping between the space of square integrable functions in the phase space \(\mathbb {R}^d\times \widehat{\mathbb {R}}^d\) and the Hilbert space of Hilbert–Schmidt operators on \(L^2(\mathbb {R}^d)\), which permits to take advantage of some well established sampling results.

Kernels of perturbed Toeplitz operators in vector-valued Hardy spaces

Recently, Liang and Partington (Integr Equ Oper Theory 92(4): 35, 2020) show that kernels of finite-rank perturbations of Toeplitz operators are nearly invariant with finite defect under the backward shift operator acting on the scalar-valued Hardy space. In this article, we provide a vectorial generalization of a result of Liang and Partington. As an immediate application, we identify the kernel of perturbed Toeplitz operator in terms of backward shift-invariant subspaces in various important cases by applying the recent theorem (see Chattopadhyay et al. in Integr Equ Oper Theory 92(6): 52, 2020, Theorem 3.5 and O’Loughlin in Complex Anal Oper Theory 14(8): 86, 2020, Theorem 3.4) in connection with nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space.