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For a given finitely generated shift invariant (FSI) subspace $cWsubset L^2(R^k)$ we obtain a simple criterion for the existence of shift generated (SG) Bessel sequences $E(cF)$ induced by finite sequences of vectors $cFin cW^n$ that have a prescribed fine structure i.e., such that the norms of the vectors in $cF$ and the spectra of $S_{E(cF)}$ is prescribed in each fiber of $text{Spec}(cW)subset T^k$. We complement this result by developing an analogue of the so-called sequences of eigensteps from finite frame theory in the context of SG Bessel sequences, that allows for a detailed description of all sequences with prescribed fine structure. Then, given $0<alpha_1leq ldotsleq alpha_n$ we characterize the finite sequences $cFincW^n$ such that $|f_i|^2=alpha_i$, for $1leq ileq n$, and such that the fine spectral structure of the shift generated Bessel sequences $E(cF)$ have minimal spread (i.e. we show the existence of optimal SG Bessel sequences with prescribed norms); in this context the spread of the spectra is measured in terms of the convex potential $P^cW_varphi$ induced by $cW$ and an arbitrary convex function $varphi:R_+rightarrow R_+$.
We are proving a Bernstein type inequality in the shift-invariant spaces of $L_2(R)$.
We introduce an extension of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in $L^2(R^k)$. We show that under a natural
Given a sequence of elements $xi={xi_n}_{nin mathbb{N}}$ of a Hilbert space, an operator $T_xi$ is defined as the operator associated to a sesquilinear form determined by $xi$. This operator is in general different to the classical frame operator but
We study the phase reconstruction of signals $f$ belonging to complex Gaussian shift-invariant spaces $V^infty(varphi)$ from spectrogram measurements $|mathcal{G}f(X)|$ where $mathcal{G}$ is the Gabor transform and $X subseteq mathbb{R}^2$. An explic
Necessary and sufficient conditions are given for density of shift-invariant subspaces of the space $mathcal{L}$ of integrable functions of bounded support with the inductive limit topology.