Structural Properties of Homomorphism Dilation Systems

Inspired by some recent development on the theory about projection valued dilations for operator valued measures or more generally bounded homomorphism dilations for bounded linear maps on Banach algebras,the authors explore a pure algebraic version of the dilation theory for linear systems acting on unital algebras and vector spaces.By introducing two natural dilation structures,namely the canonical and the universal dilation systems,they prove that every linearly minimal dilation is equivalent to a reduced homomorphism dilation of the universal dilation,and all the linearly minimal homomorphism dilations can be classified by the associated reduced subspaces contained in the kernel of synthesis operator for the universal dilation.

Stable recovery of signals from frame coefficients with erasures at unknown locations

In an earlier work, we proposed a frame-based kernel analysis approach to the problem of recovering erasures from unknown locations. The new approach led to the stability question on recovering a signal from noisy partial frame coefficients with erasures occurring at unknown locations. In this continuing work, we settle this problem by obtaining a complete characterization of frames that provide stable reconstructions. We show that an encoding frame provides a stable signal recovery from noisy partial frame coefficients at unknown locations if and only if it is totally robust with respect to erasures. We present several characterizations for either totally robust frames or almost robust frames. Based on these characterizations several explicit construction algorithms for totally robust and almost robust frames are proposed. As a consequence of the construction methods, we obtain that the probability for a randomly generated frame to be totally robust with respect to a fixed number of erasures is one.

Nonuniform sampling and approximation in Sobolev space from perturbation of the framelet system

The Sobolev space H^(■)(R^(d)),where■>d/2,is an important function space that has many applications in various areas of research.Attributed to the inertia of a measurement instrument,it is desirable in sampling theory to recover a function by its nonuniform sampling.In the present paper,based on dual framelet systems for the Sobolev space pair(H^(s)(R^(d)),H^(-s)(R^(d))),where d/2<s<■,we investigate the problem of constructing the approximations to all the functions in H^(■)(R^(d))by nonuniform sampling.We first establish the convergence rate of the framelet series in(H^(s)(R^(d)),H^(-s)(R^(d))),and then construct the framelet approximation operator that acts on the entire space H^(■)(R^(d)).We examine the stability property for the framelet approximation operator with respect to the perturbations of shift parameters,and obtain an estimate bound for the perturbation error.Our result shows that under the condition d/2<s<■,the approximation operator is robust to shift perturbations.Motivated by Hamm(2015)’s work on nonuniform sampling and approximation in the Sobolev space,we do not require the perturbation sequence to be in■^(α)(Z^(d)).Our results allow us to establish the approximation for every function in H^(■)(R^(d))by nonuniform sampling.In particular,the approximation error is robust to the jittering of the samples.