Sampling theorems and bases in a Hilbert space

A unified approach to sampling theorems for (wide sense) stationary random processes rests upon Hilbert space concepts. New results in sampling theory are obtained along the following lines: recovery of the process [chi](t) from nonperiodic samples, or when any finite number of samples are deleted; conditions for obtaining [chi](t) when only the past is sampled; a criterion for restoring [chi](t) from a finite number of consecutive samples; and a minimum mean square error estimate of [chi](t) based on any (possibly nonperiodic) set of samples.In each case, the proofs apply not only to the recovery of [chi](t), but are extended to show that (almost) arbitrary linear operations on [chi](t) can be reproduced by linear combinations of the samples. Further generality is attained by use of the spectral distribution function F([middle dot]) of [chi](t), without assuming F([middle dot]) absolutely continuous.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/32353/1/0000424.pd

Irregular and multi--channel sampling of operators

The classical sampling theorem for bandlimited functions has recently been generalized to apply to so-called bandlimited operators, that is, to operators with band-limited Kohn-Nirenberg symbols. Here, we discuss operator sampling versions of two of the most central extensions to the classical sampling theorem. In irregular operator sampling, the sampling set is not periodic with uniform distance. In multi-channel operator sampling, we obtain complete information on an operator by multiple operator sampling outputs

Frames and operators in Schatten classes

Let $T$ be a compact operator on a separable Hilbert space $H$. We show that, for $2\le p<\infty$, $T$ belongs to the Schatten class $S_p$ if and only if $\{\|Tf_n\|\}\in \ell^p$ for \emph{every} frame $\{f_n\}$ in $H$; and for $0<p\le2$, $T$ belongs to $S_p$ if and only if $\{\|Tf_n\|\}\in\ell^p$ for \emph{some} frame $\{f_n\}$ in $H$. Similar conditions are also obtained in terms of the sequence $\{\}$ and the double-indexed sequence $\{\}$.Comment: 27 page

Iterative actions of normal operators

Let $A$ be a normal operator in a Hilbert space $\mathcal{H}$, and let $\mathcal{G} \subset \mathcal{H}$ be a countable set of vectors. We investigate the relations between $A$, $\mathcal{G}$ , and $L$ that makes the system of iterations $\{A^ng: g\in \mathcal{G},\;0\leq n< L(g)\}$ complete, Bessel, a basis, or a frame for $\mathcal{H}$. The problem is motivated by the dynamical sampling problem and is connected to several topics in functional analysis, including, frame theory and spectral theory. It also has relations to topics in applied harmonic analysis including, wavelet theory and time-frequency analysis.Comment: 14 pages, 0 figure

Relationships among Interpolation Bases of Wavelet Spaces and Approximation Spaces

A multiresolution analysis is a nested chain of related approximation spaces.This nesting in turn implies relationships among interpolation bases in the approximation spaces and their derived wavelet spaces. Using these relationships, a necessary and sufficient condition is given for existence of interpolation wavelets, via analysis of the corresponding scaling functions. It is also shown that any interpolation function for an approximation space plays the role of a special type of scaling function (an interpolation scaling function) when the corresponding family of approximation spaces forms a multiresolution analysis. Based on these interpolation scaling functions, a new algorithm is proposed for constructing corresponding interpolation wavelets (when they exist in a multiresolution analysis). In simulations, our theorems are tested for several typical wavelet spaces, demonstrating our theorems for existence of interpolation wavelets and for constructing them in a general multiresolution analysis