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Register a new userConvex potentials and optimal shift generated oblique duals in shift invariant spaces
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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 normalization hypothesis, these convex potentials detect tight frames as their minimizers. We obtain a detailed spectral analysis of the frame operators of shift generated oblique duals of a fixed frame of translates. We use this result to obtain the spectral and geometrical structure of optimal shift generated oblique duals with norm restrictions, that simultaneously minimize every convex potential; we approach this problem by showing that the water-filling construction in probability spaces is optimal with respect to submajorization (within an appropriate set of functions) and by considering a non-commutative version of this construction for measurable fields of positive operators.
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We are proving a Bernstein type inequality in the shift-invariant spaces of $L_2(R)$.
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 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 explicit reconstruction formula will demonstrate that such signals can be recovered from measurements located on parallel lines in the time-frequency plane by means of a Riesz basis expansion. Moreover, connectedness assumptions on $|f|$ result in stability estimates in the situation where one aims to reconstruct $f$ on compact intervals. Driven by a recent observation that signals in Gaussian shift-invariant spaces are determined by lattice measurements [Grohs, P., Liehr, L., Injectivity of Gabor phase retrieval from lattice measurements, arXiv:2008.07238] we prove a sampling result on the stable approximation from finitely many spectrogram samples. The resulting algorithm provides a non-iterative, provably stable and convergent approximation technique. In addition, it constitutes a method of approximating signals in function spaces beyond $V^infty(varphi)$, such as Paley-Wiener spaces.
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.
We investigate dynamical properties such as topological transitivity, (sequential) hypercyclicity, and chaos for backward shift operators associated to a Schauder basis on LF-spaces. As an application, we characterize these dynamical properties for weighted generalized backward shifts on Kothe coechelon sequence spaces $k_p((v^{(m)})_{minmathbb{N}})$ in terms of the defining sequence of weights $(v^{(m)})_{minmathbb{N}}$. We further discuss several examples and show that the annihilation operator from quantum mechanics is mixing, sequentially hypercyclic, chaotic, and topologically ergodic on $mathscr{S}(mathbb{R})$.
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