arXiv subjects
Publications and source records attributed to Yurii Lyubarskii.
We obtain frame bounds estimates and the Gabor frame operator $S=S^{α,β}$ for Gabor frames generated by the Cauchy kernel. In addition we find the explicit expression for the canonical dual window for all values of the lattice parameters $α,β$, $αβ\leq 1$.
We study the frame property of the Gabor system corresponding to an irregular lattice with a single Cauchy kernel as the generating window.
We study the frame properties of the Gabor systems $$\mathfrak{G}(g;α,β):=\{e^{2πi βm x}g(x-αn)\}_{m,n\in\mathbb{Z}}.$$ In particular, we prove that for Herglotz windows $g$ such systems always form a frame for $L^2(\mathbb{R})$ if $α,β>0$, $αβ\leq1$. For general rational windows $g\in L^2(\mathbb{R})$ we prove that $\mathfrak{G}(g;α,β)$ is a frame for $L^2(\mathbb{R})$ if $0<α,β$, $αβ<1$, $αβ\not\in\mathbb{Q}$ and $\hat{g}(ξ)\neq0$, $ξ>0$, thus confirming Daubechies conjecture for this class of functions. We also discuss some related questions, in particular sampling in shift-invariant subspaces of $L^2(\mathbb{R})$.
We give a general construction of entire functions in $d$ complex variables that vanish on a lattice of the form $L = A (Z + i Z )^d$ for an invertible complex-valued matrix. As an application we exhibit a class of lattices of density >1 that fail to be a sampling set for the Bargmann-Fock space in $C ^2$. By using an equivalent real-variable formulation, we show that these lattices fail to generate a Gabor frame.
We prove sharp uniqueness results for a wide class of one-dimensional discrete evolutions. The proof is based on a construction from the theory of complex Jacobi matrices combined with growth estimates of entire functions.
We study direct and inverse scattering problem for systems of interacting particles, having web-like structure. Such systems consist of a finite number of semi-infinite chains attached to the central part formed by a finite number of particles. We assume that the semi-infinite channels are homogeneous at infinity, but the limit values of the coefficients may vary from one chain to another.
We prove that if a solution of the discrete time-dependent Schrödinger equation with bounded real potential decays fast at two distinct times then the solution is trivial. For the free Shrödinger operator and for operators with compactly supported time-independent potentials a sharp analog of the Hardy uncertainty principle is obtained, using an argument based on the theory of entire functions. Logarithmic convexity of weighted norms is employed in the case of general real-valued time-dependent bounded potentials. In the latter case the result is not optimal.
Let a sequence $Λ\subset\mathbb{C}$ be such that the corresponding system of exponential functions $\mathcal{E}(Λ):=\{e^{iλt}\}_{λ\inΛ}$ is complete and minimal in $L^2(-π,π)$ and thus each function $f\in L^2(-π,π)$ corresponds to a non-harmonic Fourier series in $\mathcal{E}(Λ)$. We prove that if the generating function $G$ of $Λ$ satisfies Muckenhoupt $(A_2)$ condition on $\mathbb{R}$, then this series admits a linear summation method. Recent results show that $(A_2)$ condition cannot be omitted.
We study the space of bandlimited Lipschitz functions in one variable. In particular we provide a geometrical description of the natural interpolating and sampling sequences for this space. We also find a description of the trace of such functions to sequences of critical density in terms of a cancellation condition.
We consider the frame property of the Gabor system G(g, α, β) = {e2πiβnt g(t - αm) : m, n \in Z} for the case of rational oversampling, i.e. α, β \in Q. A 'rational' analogue of the Ron-Shen Gramian is constructed, and prove that for any odd window function g the system G(g, α, β) does not generate a frame if αβ = (n-1)/n. Special attention is paid to the first Hermite function h_1(t) = te^(-πt^2).
We study radial behavior of harmonic functions in the unit disk belonging to the Korenblum class. We prove that functions which admit two-sided Korenblum estimate either oscillate or have slow growth along almost all radii.
Given a frame in C^n which satisfies a form of the uncertainty principle (as introduced by Candes and Tao), it is shown how to quickly convert the frame representation of every vector into a more robust Kashin's representation whose coefficients all have the smallest possible dynamic range O(1/\sqrt{n}). The information tends to spread evenly among these coefficients. As a consequence, Kashin's representations have a great power for reduction of errors in their coefficients, including coefficient losses and distortions.
We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions $H_n$. Let $h= (H_0, H_1, ..., H_n)$ be the vector of the first $n+1$ Hermite functions. We give a complete characterization of all lattices $Λ\subseteq \bR ^2$ such that the Gabor system $\{e^{2πi λ_2 t} \boh (t-λ_1): λ= (λ_1, λ_2) \in Λ\}$ is a frame for $L^2 (\bR, \bC ^{n+1})$. As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor frame and a new estimate for the lower frame bound. The main tools are growth estimates for the Weierstrass $σ$-function, a new type of interpolation problem for entire functions on the Bargmann-Fock space, and structural results about vector-valued Gabor frames.
For a scale of spaces $X$ of functions analytic in the unit disc, including the Korenblum space, and for some natural families $\mathcal E$ of uniqueness subsets for $X$, we describe minorants for $(X,\mathcal E)$, that is non-decreasing functions $M:(0,1)\to(0,\infty)$ such that $f\in X$, $E\in\mathcal E$, and $\log|f(z)|\le -M(|z|)$ on $E$ imply $f=0$. We give an application of this result to approximation by simple fractions with restrictions on the coefficients.