Shift-invariant Subspaces Research Articles

Generalized Sampling on Graphs With Subspace and Smoothness Priors

We propose a framework for generalized sampling of graph signals that parallels sampling in shift-invariant (SI) subspaces. This framework allows for arbitrary input signals, which are not constrained to be bandlimited. Furthermore, the sampling and reconstruction filters may be different. We present design methods of the correction filter that compensate for these differences and lead to closed form expressions in the graph frequency domain. In this study, we consider two priors on graph signals: The first is a subspace prior, where the signal is assumed to lie in a periodic graph spectrum (PGS) subspace. The PGS subspace is proposed as a counterpart of the SI subspace used in standard sampling theory. The second is a smoothness prior that imposes a smoothness requirement on the graph signal. We suggest the use of recovery techniques for when the recovery filter can be optimized and under a setting in which a predefined filter must be used. Sampling is performed in the graph frequency domain, which is a counterpart of "sampling by modulation" used in SI subspaces. We compare our approach with existing sampling techniques on graph signal processing. The effectiveness of the proposed generalized sampling approach is validated numerically through several experiments.

Independence of Vector-Valued Functions Associated with Kernels of Block Hankel Operators

For a block Hankel operator $$H_\Phi $$ with a matrix function symbol $$\Phi \in L^2_{M_{n\times m}}$$ , it is well-known that $$\ker H_\Phi = \Theta H^2_{\mathbb {C}^r}$$ for a natural number r and an $$m\times r$$ matrix inner function $$\Theta $$ if $$\ker H_\Phi $$ is nonempty. It will be shown that the size of the matrix inner function $$\Theta $$ associated with $$\ker H_\Phi $$ is closely related with a certain degree of independence of the column vectors of $$\Phi $$ , which will be defined in this paper. As an important application, the shape of shift invariant, or, backward shift invariant subspaces of $$H^2_{\mathbb {C}^n}$$ generated by finite elements will be studied.

Toeplitz operators on backward shift invariant subspaces of Hp

We extend results on compressed Toeplitz operators on the backward shift invariant subspaces of H2 to the context of the spaces Hp, 1<p<∞.

A Measurable Selector in Kadison’s Carpenter’s Theorem

AbstractWe show the existence of a measurable selector in Carpenter’s Theorem due to Kadison. This solves a problem posed by Jasper and the first author in an earlier work. As an application we obtain a characterization of all possible spectral functions of shift-invariant subspaces of $L^{2}(\mathbb{R}^{d})$ and Carpenter’s Theorem for type $\text{I}_{\infty }$ von Neumann algebras.

Algebraic spectral synthesis and crystal rigidity

A spectral synthesis property is obtained for closed shift-invariant subspaces of vector-valued functions on the lattice Zd. This result generalises Marcel Lefranc's 1958 theorem for scalar-valued functions. Applications are given to homogeneous systems of multivariable vector-valued discrete difference equations and to the first-order flexibility of crystallographic bar-joint frameworks.

Sampling theorem on shift invariant subspaces in mixed Lebesgue spaces

Sampling theorem is the theoretical basis of modern communication system, and is also one of the most powerful basic tools in signal processing and image processing. In this paper, we give sampling theorem on shift invariant subspaces in mixed Lebesgue spaces Lp,q(ℝd+1). When the sampling density is large enough, this sampling theorem can reconstruct exactly the signals in shift invariant subspaces. Our results improve the result in [6].

Range Spaces of Co-Analytic Toeplitz Operators

AbstractIn this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges–Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space, where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern–Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern–Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges–Rovnyak spaces and the harmonically weighted Dirichlet spaces.

The closedness of shift invariant subspaces in L^{p,q} (mathbb{R}^{d+1} )

In this paper, we consider the closedness of shift invariant subspaces in L^{p,q} (mathbb{R}^{d+1} ). We first define the shift invariant subspaces generated by the shifts of finite functions in L^{p,q} (mathbb{R}^{d+1} ). Then we give some necessary and sufficient conditions for the shift invariant subspaces in L^{p,q} (mathbb{R}^{d+1} ) to be closed. Our results improve some known results in (Aldroubi et al. in J. Fourier Anal. Appl. 7:1–21, 2001).

Approximation of Non-decaying Signals from Shift-Invariant Subspaces

In our recent work, the sampling and reconstruction of non-decaying signals, modeled as members of weighted- $$L_p$$ spaces, were shown to be stable with an appropriate choice of the generating kernel for the shift-invariant reconstruction space. In this paper, we extend the Strang–Fix theory to show that, for d-dimensional signals whose derivatives up to order L are all in some weighted- $$L_p$$ space, the weighted norm of the approximation error can be made to go down as $$O(h^L)$$ when the sampling step h tends to 0. The sufficient condition for this decay rate is that the generating kernel belongs to a particular hybrid-norm space and satisfies the Strang–Fix conditions of order L. We show that the $$O(h^L)$$ behavior of the error is attainable for both approximation schemes using projection (when the signal is prefiltered with the dual kernel) and interpolation (when a prefilter is unavailable). The requirement on the signal for the interpolation method, however, is slightly more stringent than that of the projection because we need to increase the smoothness of the signal by a margin of $$d/p+\varepsilon $$ , for arbitrary $$\varepsilon >0$$ . This extra amount of derivatives is used to make sure that the direct sampling is stable.

A free interpolation problem for a subspace of H∞

Given an inner function $\theta$, the associated star-invariant subspace $K^\infty_\theta$ is formed by the functions $f\in H^\infty$ that annihilate (with respect to the usual pairing) the shift-invariant subspace $\theta H^1$ of the Hardy space $H^1$. Assuming that $B$ is an interpolating Blaschke product with zeros $\{a_j\}$, we characterize the traces of functions from $K^\infty_B$ on the sequence $\{a_j\}$. The trace space that arises is, in general, non-ideal (i.e., the sequences $\{w_j\}$ belonging to it admit no nice description in terms of the size of $|w_j|$), but we do point out explicit -- and sharp -- size conditions on $|w_j|$ which make it possible to solve the interpolation problem $f(a_j)=w_j$ ($j=1,2,\dots$) with a function $f\in K^\infty_B$.

On a class of shift-invariant subspaces of the Drury-Arveson space

Abstract In the Drury-Arveson space, we consider the subspace of functions whose Taylor coefficients are supported in a set Y⊂ ℕd with the property that ℕ\X + ej ⊂ ℕ\X for all j = 1, . . . , d. This is an easy example of shift-invariant subspace, which can be considered as a RKHS in is own right, with a kernel that can be explicitly calculated for specific choices of X. Every such a space can be seen as an intersection of kernels of Hankel operators with explicit symbols. Finally, this is the right space on which Drury’s inequality can be optimally adapted to a sub-family of the commuting and contractive operators originally considered by Drury.

Sampling Formulas Involving Differences in Shift-Invariant Subspaces: A Unified Approach

ABSTRACTSuccessive differences on a sequence of data help discover some smoothness features of this data. This was one of the main reasons for rewriting the classical interpolation formula in terms of such data differences. The aim of this paper is to mimic them to a sequence of regular samples of a function in a shift-invariant subspace allowing its stable recovery. A suitable expression for the functions in the shift-invariant subspace by an isomorphism with the L2(0,1) space is the key to identify the simple pattern followed by the dual Riesz bases involved in the derived formulas. The paper contains examples illustrating different non-exhaustive situations including also the two-dimensional case.

Asymmetric Truncated Toeplitz Operators of Rank One

A truncated Toeplitz operator is a compression of the multiplicationoperator to a backward shift invariant subspace of the Hardy space H^2. Anasymmetric truncated Toeplitz operator is a compression of the multiplication operator that acts between two different backward shift invariant subspaces of H^2. All rank-one truncated Toeplitz operators have been described by Sarason. Here, we characterize all rank-one asymmetric truncated Toeplitz operators. This completes the description given by Łanucha for asymmetric truncated Toeplitz operators on finite-dimensional backward shift invariant subspaces.

Weyl transform and Weyl multipliers associated with locally compact abelian groups

In this paper, we extend to the context of locally compact abelian groups, the notion of Weyl transform. We also define and characterize Weyl multipliers for certain function spaces. A dual space characterization is provided for the space of Weyl multipliers on \(L^p\) spaces. Finally we also study the twisted shift-invariant subspaces.

Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue spaces [formula omitted

In this paper, we study the nonuniform sampling and reconstruction problem in shift-invariant subspaces of mixed Lebesgue spaces. We first show that shift-invariant subspaces in mixed Lebesgue spaces Lp,q(Rd+1) can be well-defined. Then we propose that the sampling problem in shift-invariant subspaces of mixed Lebesgue spaces is well-posed. At last, the nonuniform samples {f(xj,yk):k,j∈J} of a function f belonging to a shift-invariant subspace of mixed Lebesgue spaces are proposed, and we give a fast reconstruction algorithm that allows exact reconstruction of f as long as the sampling set X={(xj,yk):k,j∈J} is sufficiently dense.

Adjoint of the Toeplitz operator with the singular inner function

Let ψδ1 be the singular inner function associated with the unit point mass at z=1. We shall give an infinite dimensional closed subspace M of kerTψδ1⁎, the kernel of the adjoint of the Toeplitz operator Tψδ1, satisfying that for every nonzero v in M, the smallest backward shift invariant subspace of H2 containing v coincides with kerTψδ1⁎.

Kernels of Toeplitz operators on the Hardy space over the bidisk

We study the kernels of Toeplitz operators on the Hardy space on the bidisk. We first give a sufficient condition for a general symbol to be antiholomorphic under the assumption that the kernel of the corresponding Toeplitz operator contains a backward shift invariant subspace. As an application, we construct an invariant subspace whose its orthogonal complement can not be the kernel of any Toeplitz operator. Also, we give a characterization on a Toeplitz operator for which the orthogonal complement of its kernel is generated by certain inner functions. Finally, we describe all backward shift invariant subspaces which are in the kernels of Toeplitz operators with homogeneous type symbols. As an application, we show that there is a Toeplitz operator for which its kernel has dimension of any given integer and the orthogonal complement of the kernel is generated by two functions. Our result shows that there are higher dimensional phenomena.

Krylov--Schur-Type Restarts for the Two-Sided Arnoldi Method

We consider the two-sided Arnoldi method and propose a two-sided Krylov--Schur-type restarting method. We discuss the restart for standard Rayleigh--Ritz extraction as well as harmonic Rayleigh--Ritz extraction. Additionally, we provide error bounds for Ritz values and Ritz vectors in the context of oblique projections and present generalizations of, e.g., the Bauer--Fike theorem and Saad's theorem. Applications of the two-sided Krylov--Schur method include the simultaneous computation of left and right eigenvectors and the computation of eigenvalue condition numbers. We demonstrate how the method can be used to find the least sensitive eigenvalues of a nonnormal matrix and how to approximate pseudospectra by using left and right shift-invariant subspaces. The results demonstrate that significant improvements in quality can be obtained over approximations with the (one-sided) Krylov--Schur method.

A general Beurling–Helson–Lowdenslager theorem on the disk

The classical Beurling–Helson–Lowdenslager theorem characterizes the shift-invariant subspaces of the Hardy space H2 and of the Lebesgue space L2. In this paper, which is self-contained, we define a very general class of norms α and define spaces Hα and Lα. We then extend the Beurling–Helson–Lowdenslager invariant subspace theorem. The idea of the proof is new and quite simple; most of the details involve extending basic well-known ‖⋅‖p-results for our more general norms.

A Local Weighted Average Sampling and Reconstruction Theorem over Shift Invariant Subspaces

We analyse a local average sampling and reconstruction problem over some shift invariant subspaces of \({L^{2}({\mathbb{R}})}\). We present certain necessary and sufficient conditions under which there is an average sampling expansion for both uniform sampling points and non-uniform sampling points.