Generalized Shift-Invariant Systems

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.