arXiv subjects
Publications and source records attributed to Yurii Belov.
We prove that resonances of the Schr\"{o}dinger operator with compactly supported potential can contain arbitrary subset of the angle $\{z: -\text{Im} z > C |\text{Re} z|\}$ that satisfies Blaschke condition. We also establish sufficient conditions for the subsets of wider domains.
We find sufficient conditions on a compactly supported function $g$, $\supp g = [a,b]$ which guarantee that the Gabor system $$\mathcal{G}(g;\alpha,\beta)=\{e^{2\pi i \beta m x}g(x-\alpha n)\}_{m,n\in\mathbb{Z}}$$ is a frame for all $\alpha < b-a, \alpha\beta < 1, \alpha\beta \notin\Q$. These conditions are on one hand satisfied by almost all such functions, and on the other hand are explicit enough that we can give many concrete examples of the functions $g$ which give us a frame e.g. $g(x) = \exp(\frac{1}{x^4-1})\chi_{(-1,1)}(x)$.
We consider the sampling problem for two-sided small Fock spaces $\mathcal{F}^p_{\alpha}$, for the full range $0 < p \le \infty$. We establish a geometric description of shift-invariant sampling sequences, i.e., sequences $\Lambda$ such that $c \Lambda$ is sampling for all $c \in \mathbb{C} \setminus \{ 0 \}$.
Let $g\in L^2(\mathbb{R})$ be a strictly decreasing continuous function supported on $\mathbb{R}_+$ such that for all $t > 0$ we have $g(x+t)\le q(t)g(x)$ for some $q(t)<1$. We prove that the Gabor system $$\mathcal{G}(g;\alpha,\beta):=\{g_{m,n}\}_{m,n\in\mathbb{Z}}=\{e^{2\pi i \beta m x}g(x-\alpha n)\}_{m,n\in\mathbb{Z}}$$ always forms a frame in $L^2(\mathbb{R})$ for all lattice parameters $\alpha$,$\beta$, $\alpha\beta\leq 1$.
We establish a relation between the approximation in $L^2[-\pi,\pi]$ by exponentials with the set of frequencies of Beurling--Malliavin density less than $1$ and the meromorphic interpolation at $\mathbb Z$. Furthermore, we show that typical $L^2[-\pi,\pi]$ functions admit such an approximation.
We study Gabor frames in the case when the window function is of hyperbolic secant type, i.e., $g(x) = (e^{ax}+e^{-bx})^{-1}$, ${\rm Re}\,a, {\rm Re}\,b>0$. A criterion for half-irregular sampling is obtained: for a separated $\Lambda\subset\mathbb{R}$ the Gabor system $\mathcal{G}(g, \Lambda \times \alpha\Z)$ is a frame in $L^2(\R)$ if and only if $D^-(\Lambda) >\alpha$ where $D^-(\Lambda)$ is the usual (Beurling) lower density of $\Lambda$. This extends a result by Gr\"ochenig, Romero, and St\"ockler which applies to the case of a standard hyperbolic secant. Also, a full description of complete interpolating sequences for the shift-invariant space generated by $g$ is given.
We prove that frame set $\mathcal{F}_g$ for imaginary shift of sinc-function $$g(t)=\frac{\sinπb(t-iw)}{t-iw}, \quad b,w\in\mathbb{R}\setminus\{0\}$$ can be described as $\mathcal{F}_g=\{(α,β): αβ\leq 1, β\leq|b|\}.$ \\ In addition, we prove that $\mathcal{F}_g=\{(α,β): αβ\leq 1 \}$ for window functions $g$ of the form $\frac{1}{t-iw}(1-\sum\limits_{k=1}^{\infty}a_ke^{2πi b_k t})$, such that $\sum_{k\geq 1}|a_k|e^{2π|w|b_k}<1$, $wb_k<0$.
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 give sufficient conditions for the exponential system to be a Riesz basis in $L^2(E)$, where $E$ is a union of two intervals. We show that these conditions are close to be necessary. In addition, we demonstrate ``extra point effect'' for such systems, i.e. it may happen that the Riesz basis in $L^2(E)$ differs by one point from the Riesz basis on an interval.
We study the properties of a system biorthogonal to a complete and minimal system of exponentials in $L^2(E)$, where $E$ is a finite union of intervals, and show that in the case when $E$ is a union of two or three intervals the biorthogonal system is also complete.
We study the class of discrete measures in the complex plain with the following property: up to a finite number, all zeros of any Cauchy transform of the measure (with $\ell^2$-data) are localized near the support of the measure. We find several equivalent forms of this property and prove that the parts of the support attracting zeros of Cauchy transforms are ordered by inclusion modulo finite sets.
We study the indivisible intervals and the monotonicity of the growth of the exponential type in the chains of de Branges subspaces in terms of the spectral measure. We prove that for spectral measures supported on $\mathbb Z$, there exist at most two subspaces of the same type, which then bound an indivisible interval. Furthermore, in this case, we study possible locations of the indivisible intervals.
We give a full description of complete interpolating sequences for the shift-invariant space generated by the Gaussian. As a consequence, we rederive the known density conditions for sampling and interpolation.
We give a negative answer to the Newman--Shapiro problem on weighted approximation for entire functions formulated in 1966 and motivated by the theory of operators on the Fock space. There exists a function in the Fock space such that its exponential multiples do not approximate some entire multiples in the space. Furthermore, we establish several positive results under different restrictions on the function in question.
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 study hereditary completeness of systems of exponentials on an interval such that the corresponding generating function $G$ is small outside of a lacunary sequence of intervals $I_k$. We show that, under some technical conditions, an exponential system is hereditarily complete if and only if the logarithmic length of the union of these intervals is infinite, i.e., $\sum_k\int_{I_k} \frac{dx}{1+|x|}=\infty$.
We study the structure of the backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces $\mathcal{F}_W^p$, whose weight $W$ is not necessarily radial. We show that in the spaces $\mathcal{F}_W^p$ which contain the polynomials as a dense subspace (in particular, in the radial case) all nontrivial backward shift invariant subspaces are of the form $\mathcal{P}_n$, i.e., finite dimensional subspaces consisting of polynomials of degree at most $n$. In general, the structure of the nearly invariant subspaces is more complicated. In the case of spaces of slow growth (up to zero exponential type) we establish an analogue of de Branges' Ordering Theorem. We then construct examples which show that the result fails for general Fock-type spaces of larger growth.