Generalized Shift-Invariant Systems |
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Authors: | Amos Ron Zuowei Shen |
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Institution: | (1) Computer Sciences Department, University of Wisconsin-Madison, 1210 West Dayton, Madison, WI 53706, USA |
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Abstract: | A countable collection $X$ of functions in $L_2(\mbox{\footnotesize\bf
R})$ is said to be a Bessel system if the associated
analysis operator
$$
\txs{X}:L_2(\mbox{\smallbf R}^d)\to \ell_2(X) : f\mapsto (\inpro{f,x})_{x\in X}
$$
is well-defined and bounded. A Bessel system is a fundamental frame if
$\txs{X}$ is injective and its range is closed.
This paper considers the above two properties for a generalized
shift-invariant system $X$. By definition, such a system has the form
$$
X=\bigcup_{j\in J} Y_j,
$$
where each $Y_j$ is a shift-invariant system (i.e., is comprised of lattice
translates of some function(s)) and $J$ is a countable (or finite) index set.
The definition is general enough to include wavelet systems, shift-invariant
systems, Gabor systems, and many variations of wavelet systems such as
quasi-affine ones and nonstationary ones.
The main theme of this paper is the fiberization of $\txs{X}$,
which allows one to study the frame and Bessel properties of $X$ via the spectral
properties of a collection of finite-order Hermitian nonnegative matrices. |
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Keywords: | Dual Gramian Frames Wavelets Shift-invariant
systems Quasi-affine system Oversampling |
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