Summation method in optimal control problem with delay
We describe a summation procedure for the construction of the optimal solution in null controllability problem for differential equation with distributed delay
math.OC↗
arXiv subjects
Yu. Lyubarskii
Publications and source records attributed to Yu. Lyubarskii.
On Gabor orthonormal bases over finite prime fields
We study Gabor orthonormal windows in $L^2({\Bbb Z}_p^d)$ for translation and modulation sets $A$ and $B$, respectively, where $p$ is prime and $d\geq 2$. We prove that for a set $E\subset \Bbb Z_p^d$, the indicator function $1_E$ is a Gabor window if and only if $E$ tiles and is spectral. Moreover, we prove that for any function $g:\Bbb Z_p^d\to \Bbb C$ with support $E$, if the size of $E$ coincides with the size of the modulation set $B$ or if $g$ is positive, then $g$ is a unimodular function, i.e., $|g|=c1_E$, for some constant $c>0$, and $E$ tiles and is spectral. We also prove the existence of a Gabor window $g$ with full support where neither $|g|$ nor $|\hat g|$ is an indicator function and $|B|<<p^d$. We conclude the paper with an example and open questions.
math.CA↗
Trace ideal criteria for embeddings and composition operators on model spaces
Let $K_θ$ be a model space generated by an inner function $θ$. We study the Schatten class membership of embeddings $I : K_θ\to L^2(μ)$, $μ$ a positive measure, and of composition operators $C_ϕ:K_θ\to H^2(\mathbb D)$ with a holomprphic function $ϕ:\mathbb D\rightarrow \mathbb D$. In the case of one-component inner functions $θ$ we show that the problem can be reduced to the study of natural extensions of $I$ and $C_ϕ$ to the Hardy-Smirnov space $E^2(D)$ in some domain $D\supset \mathbb D$. In particular, we obtain a characterization of Schatten membership of $C_ϕ$ in terms of Nevanlinna counting function. By example this characterization does not hold true for general $ϕ$.
math.FA↗
Frame Constants of Gabor Frames near the Critical Density
We consider Gabor frames generated by a Gaussian function and describe the behavior of the frame constants as the density of the lattice approaches the critical value.
math.CV↗
Riesz bases of reproducing kernels in Fock type spaces
In a scale of Fock spaces $\mathcal F_φ$ with radial weights $φ$ we study the existence of Riesz bases of (normalized) reproducing kernels. We prove that these spaces possess such bases if and only if $φ(x)$ grows at most like $(\log x)^2$.
math.CV↗
Radial growth of functions from the Korenblum space
We study radial behavior of analytic and harmonic functions, which admit a certain majorant in the unit disk. We prove that extremal growth or decay may occur only along small sets of radii and give precise estimates of these exceptional sets.
math.CA↗
A Hilbert Lemniscate Theorem in C^2
For a regular, compact, polynomially convex circled set K in C^2, we construct a sequence of pairs {P_n,Q_n} of homogeneous polynomials in two variables with deg P_n = deg Q_n = n such that the sets K_n: = {(z,w) \in C^2 : |P_n(z,w)| \leq 1, |Q_n(z,w)| \leq 1} approximate K and the normalized counting measures {μ_n} associated to the finite set {P_n = Q_n = 1} converge to the pluripotential-theoretic Monge-Ampere measure for K. The key ingredient is an approximation theorem for subharmonic functions of logarithmic growth in one complex variable.
math.CV↗