arXiv subjects
Publications and source records attributed to Yurii Belov.
We study the upper and the lower densities of complete and minimal Gabor Gaussians systems. In contrast to the classical lattice case when they are both equal to $1$, we prove that the lower density may reach $0$ while the upper density may vary at least from $\frac{1}π$ to $e$. In the case when the upper density exceeds $1$, we establish a sharp inequality relating the upper and the lower densities.
We show that there exist complete and minimal systems of time-frequency shifts of Gaussians in $L^2(\mathbb{R})$ which are not strong Markushevich basis (do not admit the spectral synthesis). In particular, it implies that there is no linear summation method for general Gaussian Gabor expansions. On the other hand we prove that the spectral synthesis for such Gabor systems holds up to one dimensional defect.
We establish the existence of Nevanlinna domains with large boundaries. In particular, these domains can have boundaries of positive planar measure. The sets of accessible points can be of any Hausdorff dimension between $1$ and $2$. As a quantitative counterpart of these results, we construct rational functions univalent in the unit disc with extremely long boundaries for a given amount of poles.
We describe differentiation-invariant subspaces of $C^\infty(a,b)$ which admit spectral synthesis. This gives a complete answer to a question posed by A.~Aleman and B.~Korenblum. It turns out that this problem is related to a classical problem of approximation by polynomials on the real line. We will depict an intriguing connection between these problems and the theory of de Branges spaces.
We extend some results of M.G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustrating sharpness of the obtained results are given.
Motivated by a problem in approximation theory, we find a necessary and sufficient condition for a model (backward shift invariant) subspace $K_\varTheta = H^2\ominus \varTheta H^2$ of the Hardy space $H^2$ to contain a bounded univalent function.
We prove that for every radial weighted Fock space, the system biorthogonal to a complete and minimal system of reproducing kernels is also complete under very mild regularity assumptions on the weight. This result generalizes a theorem by Young on reproducing kernels in the Paley--Wiener space and a recent result of Belov for the classical Bargmann--Segal--Fock space.
We describe the radial Fock type spaces which possess Riesz bases of normalized reproducing kernels and which are (are not) isomorphic to de Branges spaces in terms of the weight functions.
We characterize the Hermite-Biehler (de Branges) functions $E$ which correspond to Shroedinger operators with $L^2$ potential on the finite interval. From this characterization one can easily deduce a recent theorem by Horvath. We also obtain a result about location of resonances.
We give a simple and straightforward proof of the Gap Theorem for separated sequences by A. Poltoratski and M. Mitkovski using the Beurling--Malliavin formula for the radius of completeness.
We solve completely the spectral synthesis problem for reproducing kernels in the de Branges spaces $\mathcal{H}(E)$. Namely, we describe the de Branges spaces $\mathcal{H}(E)$ such that all $M$-bases of reproducing kernels (i.e., complete and minimal systems $\{k_λ\}_{λ\inΛ}$ with complete biorthogonal $\{g_λ\}_{λ\inΛ}$) are strong $M$-bases (i.e., every mixed system $\{k_λ\}_{λ\inΛ\setminus\tilde Λ} \cup\{g_λ\}_{λ\in \tilde Λ}$ is also complete). Surprisingly this property takes place only for two essentially different classes of de Branges spaces: spaces with finite spectral measure and spaces which are isomorphic to Fock-type spaces of entire functions. The first class goes back to de Branges himself, the second class appeared in a recent work of A. Borichev and Yu. Lyubarskii. Moreover, we are able to give a complete characterisation of this second class in terms of the spectral data for $\mathcal{H}(E)$. In addition, we obtain some results about possible codimension of mixed systems for a fixed de Branges space $\mathcal{H}(E)$, and prove that any minimal system of reproducing kernels in $\mathcal{H}(E)$ is contained in an exact system of reproducing kernels.
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 prove that any complete and minimal Gabor system of Gaussians is a Markushevich basis in $L^2(\mathbb{R})$.
We prove that any incomplete system of complex exponentials $\{e^{iλ_n t}\}$ in $L^2(-π,π)$ is a subset of some complete and minimal system of exponentials. In addition, we prove analogous statement for systems of reproducing kernels in de Branges spaces.
We study the localization of zeros of Cauchy transforms of discrete measures on the real line. This question is motivated by the theory of canonical systems of differential equations. In particular, we prove that the spaces of Cauchy transforms having the localization property are in one-to-one correspondence with the canonical systems of special type, namely, those whose Hamiltonians consist only of indivisible intervals accumulating on the left. Various aspects of the localization phenomena are studied in details. Connections with the density of polynomials and other topics in analysis are discussed.
Let $L$ be a proper differentiation invariant subspace of $C^\infty(a,b)$ such that the restriction operator $\frac{d}{dx}\bigl{|}_L$ has a discrete spectrum $Λ$ (counting with multiplicities). We prove that $L$ is spanned by functions vanishing outside some closed interval $I\subset(a,b)$ and monomial exponentials $x^ke^{λx}$ corresponding to $Λ$ if its density does not exceed the critical value $\frac{|I|}{2π}$, and moreover, we show that the result is not necessarily true when the density of $Λ$ equals the critical value. This answers a question posed by the first author and B. Korenblum. Finally, if the residual part of $L$ is trivial, then $L$ is spanned by the monomial exponentials it contains.
Let $ω$ be a non-negative function on $\mathbb{R}$. We are looking for a non-zero $f$ from a given space of entire functions $X$ satisfying $$(a) \quad|f|\leq ω\text{\quad or\quad(b)}\quad |f|\asympω.$$ The classical Beurling--Malliavin Multiplier Theorem corresponds to $(a)$ and the classical Paley--Wiener space as $X$. We survey recent results for the case when $X$ is a de Branges space $\he$. Numerous answers mainly depend on the behaviour of the phase function of the generating function $E$.
We survey recent results concerning the hereditary completeness of some special systems of functions and the spectral synthesis problem for a related class of linear operators. We present a solution of the spectral synthesis problem for systems of exponentials in $L^2(-π, π)$. Analogous results are obtained for the systems of reproducing kernels in the de Branges spaces of entire functions. We also apply these results (via a functional model) to the spectral theory of rank one perturbations of compact self-adjoint operators.