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Alexander Borichev

Publications and source records attributed to Alexander Borichev.

At least 55 records · Page 3Linked to original sources
arXiv2012

A restricted shift completeness problem

We solve a problem about the orthogonal complement of the space spanned by restricted shifts of functions in $L^2[0,1]$ posed by M.Carlsson and C.Sundberg.

math.CV↗
arXiv2012

Hereditary completeness for systems of exponentials and reproducing kernels

We solve the spectral synthesis problem for exponential systems on an interval. Namely, we prove that any complete and minimal system of exponentials $\{e^{iλ_n t}\}$ in $L^2(-a,a)$ is hereditarily complete up to a one-dimensional defect. This means that there is at most one (up to a constant factor) function $f$ which is orthogonal to all the summands in its formal Fourier series $\sum_n (f,\tilde e_n) e^{iλ_n t}$, where $\{\tilde e_n\}$ is the system biorthogonal to $\{e^{iλ_n t}\}$. However, this one-dimensional defect is possible and, thus, there exist nonhereditarily complete exponential systems. Analogous results are obtained for systems of reproducing kernels in de Branges spaces. For a wide class of de Branges spaces we construct nonhereditarily complete systems of reproducing kernels, thus answering a question posed by N. Nikolski.

math.CV↗
arXiv2012

Generalized Carleson--Newman inner functions

We study a class of inner functions introduced by Gorkin, Mortini, and Nikolski, and motivated by Banach algebras and functional calculus applications. Answering their question, we produce a singular function that cannot be multiplied into this generalized Carleson-Newman class of inner functions (the class of functions satisfying the so called weak embedding property). Furthermore, we give elementary proofs for some results on these classes obtained earlier using arguments related to the Gelfand representation of $H^\infty$.

math.CV↗
arXiv2011

Remarks on Eisenstein

We obtain a fully explicit quantitative version of the Eisenstein theorem on algebraic power series which is more suitable for certain applications than the existing version due to Dwork, Robba, Schmidt and van der Poorten. We also treat ramified series and Laurent series, and we demonstrate some applications; for instance, we estimate the discriminant of the number field generated by the coefficients.

math.NT↗
arXiv2011

Weighted exponential approximation and non-classical orthogonal spectral measures

A long-standing open problem in harmonic analysis is: given a non-negative measure $μ$ on $\mathbb R$, find the infimal width of frequencies needed to approximate any function in $L^2(μ)$. We consider this problem in the "perturbative regime", and characterize asymptotic smallness of perturbations of measures which do not change that infimal width. Then we apply this result to show that there are no local restrictions on the structure of orthogonal spectral measures of one-dimensional Schrodinger operators on a finite interval. This answers a question raised by V.A.Marchenko.

math.CA↗
arXiv2011

On Burkholder function for orthogonal martingales and zeros of Legendre polynomials

Burkholder obtained a sharp estimate of $\E|W|^p$ via $\E|Z|^p$, where $W$ is a martingale transform of $Z$, or, in other words, for martingales $W$ differentially subordinated to martingales $Z$. His result is that $\E|W|^p\le (p^*-1)^p\E|Z|^p$, where $p^* =\max (p, \frac{p}{p-1})$. What happens if the martingales have an extra property of being orthogonal martingales? This property is an analog (for martingales) of the Cauchy-Riemann equation for functions, and it naturally appears from a problem on singular integrals (see the references at the end of Section~1). We establish here that in this case the constant is quite different. Actually, $\E|W|^p\le (\frac{1+z_p}{1-z_p})^p\E|Z|^p$, $p\ge 2$, where $z_p$ is a specific zero of a certain solution of a Legendre ODE. We also prove the sharpness of this estimate. Asymptotically, $(1+z_p)/(1-z_p)=(4j^{-2}_0+o(1))p$, $p\to\infty$, where $j_0$ is the first positive zero of the Bessel function of zero order. This connection with zeros of special functions (and orthogonal polynomials for $p=n(n+1)$) is rather unexpected.

math.CA↗
arXiv2011

Subordination by orthogonal martingales in $L^{p}$ and zeros of Laguerre polynomials

In this paper we address the question of finding the best $L^p$-norm constant for martingale transforms with one-sided orthogonality. We consider two martingales on a probability space with filtration $\mathcal{B}$ generated by a two-dimensional Brownian motion $B_t$. One is differentially subordinated to the other. Here we find the sharp estimate for subordinate martingales if the subordinated martingale is orthogonal and $1 2$, but the orthogonal martingale is a subordinator. The answers are given in terms of zeros of Laguerre polynomials. As an application of our sharp constant we obtain a new estimate for the norm of theAhlfors--Beurling operator. We estimate it as $1.3922(p-1)$ asymptotically for large $p$.

math.PR↗
arXiv2010

Stable and Norm-stable Invariant Subspaces

We prove that if T is an operator on an infinite-dimensional Hilbert space whose spectrum and essential spectrum are both connected and whose Fredholm index is only 0 or 1, then the only nontrivial norm-stable invariant subspaces of T are the finite-dimensional ones. We also characterize norm-stable invariant subspaces of any weighted unilateral shift operator. We show that quasianalytic shift operators are points of norm continuity of the lattice of the invariant subspaces. We also provide a necessary condition for strongly stable invariant subspaces for certain operators.

math.FA↗
arXiv2009

Optimal polynomial decay of functions and operator semigroups

We characterize the polynomial decay of orbits of Hilbert space $C_0$-semigroups in resolvent terms. We also show that results of the same type for general Banach space semigroups and functions obtained recently in the paper by C.J.K.Batty and T.Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces (J. Evol. Eq. 2008), are sharp. This settles a conjecture posed in the paper by C.J.K.Batty and T.Duyckaerts.

math.FA↗
arXiv2009

Estimates of holomorphic functions in zero-free domains

We study functions f(z) holomorphic in the upper half plane and having no zeros when the imaginary part of z is between 0 and 1, and we obtain a lower bound for the modulus of f(z) in this strip. In our analysis we deal with scalar functions f(z) as well as with operator valued holomorphic functions I+A(z) assuming that A(z) is a trace class operator in the upper half plane and I+A(z) is invertible in the same strip and is unitary on the real line.

math.CV↗
arXiv2006

Majorants of meromorphic functions with fixed poles

Let $B$ be a meromorphic Blaschke product in the upper half-plane with zeros $z_n$ and let $K_B=H^2\ominus BH^2$ be the associated model subspace of the Hardy class. In other words, $K_B$ is the space of square summable meromorphic functions with the poles at the points $\bar z_n$. A nonnegative function $w$ on the real line is said to be an admissible majorant for $K_B$ if there is a non-zero function $f\in K_B$ such that $|f|\le w$ a.e. on $\mathbb{R}$. We study the relations between the distribution of the zeros of a Blaschke product $B$ and the class of admissible majorants for the space $K_B$.

math.CV↗
arXiv2005

Uniqueness theorems for Korenblum type spaces

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.

math.CV↗
arXiv2003

Harmonic and superharmonic majorants on the disk

We prove that a positive function on the unit disk admits a harmonic majorant if and only if a certain logarithmic Lipschitz upper envelope of it (relevant because of the Harnack inequality) admits a superharmonic majorant. We discuss the logarithmic Lipschitz regularity of this superharmonic majorant, and show that in general it cannot be better than that of the Poisson kernel. We provide examples to show that mere superharmonicity of the data does not help with the problem of existence of a harmonic majorant.

math.CV↗
arXiv2003

Lower bounds for quasianalytic functions, II. The Bernstein quasianalytic functions

Let F be a class of functions with the uniqueness property: if a function f in F vanishes on a set of positive measure, then f is the zero function. In many instances, we would like to have a quantitative version of this property, e.g. a lower bound for |f| outside a small exceptional set. Such estimates are well-known and useful for polynomials, complex- and real-analytic functions, exponential polynomials. In this work we prove similar results for the Denjoy-Carleman and the Bernstein classes of quasianalytic functions. In the first part (this arXiv:math.CA/0208233), we considered quasianalytically smooth functions. Here, we deal with classes of functions characterized by exponentially fast approximation by polynomials whose degrees belong to a given very lacunar sequence. We also prove the polynomial spreading lemma and a comparison lemma which are of a certain interest on their own.

math.CA↗