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Sep 21, 2013
09/13

by
Anton Baranov; Yurii Belov; Alexander Borichev

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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.

Source: http://arxiv.org/abs/1204.4648v1

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Sep 20, 2013
09/13

by
Alexander Borichev; Prabhu Janakiraman; Alexander Volberg

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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...

Source: http://arxiv.org/abs/1002.2314v3

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27

Jun 30, 2018
06/18

by
Charles Batty; Alexander Borichev; Yuri Tomilov

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We prove $L^p$-analogues of the classical tauberian theorem of Ingham and Karamata, and its variations giving rates of decay. These results are applied to derive $L^p$-decay of operator families arising in the study of the decay of energy for damped wave equations and local energy for wave equations in exterior domains. By constructing some examples of critical behaviour we show that the $L^p$-rates of decay obtained in this way are best possible under our assumptions.

Topics: Complex Variables, Functional Analysis, Mathematics, Analysis of PDEs, Dynamical Systems

Source: http://arxiv.org/abs/1403.6084

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38

Sep 18, 2013
09/13

by
Alexander Borichev; Haakan Hedenmalm

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To address the uniqueness issues associated with the Dirichlet problem for the $N$-harmonic equation on the unit disk $\D$ in the plane, we investigate the $L^p$ integrability of $N$-harmonic functions with respect to the standard weights $(1-|z|^2)^{\alpha}$. The question at hand is the following. If $u$ solves $\Delta^N u=0$ in $\D$, where $\Delta$ stands for the Laplacian, and [\int_\D|u(z)|^p (1-|z|^2)^{\alpha}\diff A(z)

Source: http://arxiv.org/abs/1211.5088v3

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Sep 24, 2013
09/13

by
Yuri Bilu; Alexander Borichev

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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.

Source: http://arxiv.org/abs/1112.2290v2

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58

Sep 20, 2013
09/13

by
Alexander Borichev; Fedor Nazarov; Mikhail Sodin

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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...

Source: http://arxiv.org/abs/math/0301217v2

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3.0

Jun 30, 2018
06/18

by
Alexander Borichev; Mikhail Sodin; Benjamin Weiss

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We will discuss a somewhat striking spectral property of finitely valued stationary processes on Z that says that if the spectral measure of the process has a gap then the process is periodic. We will give some extensions of this result and raise several related questions.

Topics: Classical Analysis and ODEs, Probability, Mathematics

Source: http://arxiv.org/abs/1701.03407

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2.0

Jun 28, 2018
06/18

by
Anton Baranov; Yurii Belov; Alexander Borichev

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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.

Topics: Functional Analysis, Complex Variables, Mathematics

Source: http://arxiv.org/abs/1512.09001

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Sep 23, 2013
09/13

by
Alexander Borichev; Grigori Rozenblum

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We consider Toeplitz operators in the Fock space, under rather general conditions imposed on the symbols. It is proved that if the operator has finite rank and the symbol is a function then the operator and the symbol should be zero. The method of proving is different from the one used previously for finite rank theorems, and it enables one to get rid of the compact support condition for symbols imposed previously.

Source: http://arxiv.org/abs/1303.2996v1

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Sep 17, 2013
09/13

by
Alexander Borichev; Yuri Tomilov

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We obtain new uniqueness theorems for harmonic functions defined on the unit disc or in the half plane. These results are applied to obtain new resolvent descriptions of spectral subspaces of polynomially bounded groups of operators on Banach spaces.

Source: http://arxiv.org/abs/1003.2805v1

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Sep 22, 2013
09/13

by
Alexander Borichev; Mikhail Sodin

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A long-standing open problem in harmonic analysis is: given a non-negative measure $\mu$ on $\mathbb R$, find the infimal width of frequencies needed to approximate any function in $L^2(\mu)$. 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...

Source: http://arxiv.org/abs/1004.1795v3

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Sep 19, 2013
09/13

by
Alexander Borichev; Hakan Hedenmalm; Alexander Volberg

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In a wide class of weighted Bergman spaces, we construct invertible non-cyclic elements. These are then used to produce z-invariant subspaces of index higher than one. In addition, these elements generate nontrivial bilaterally invariant subspaces in anti-symmetrically weighted Hilbert spaces of sequences.

Source: http://arxiv.org/abs/math/0304318v1

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Sep 22, 2013
09/13

by
Alexander Borichev; Vesselin Petkov

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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.

Source: http://arxiv.org/abs/0906.1827v1

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4.0

Jun 30, 2018
06/18

by
Alexander Borichev; Alon Nishry; Mikhail Sodin

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We study the influence of the multipliers $\xi (n)$ on the angular distribution of zeroes of the Taylor series \[ F_\xi (z) = \sum_{n\ge 0} \xi (n) \frac{z^n}{n!}\,. \] We show that the distribution of zeroes of $ F_\xi $ is governed by certain autocorrelations of the sequence $ \xi $. Using this guiding principle, we consider several examples of random and pseudo-random sequences $\xi$ and, in particular, answer some questions posed by Chen and Littlewood in 1967. As a by-product we show that...

Topics: Complex Variables, Probability, Mathematics

Source: http://arxiv.org/abs/1409.2736

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Sep 23, 2013
09/13

by
Anton Baranov; Yurii Belov; Alexander Borichev; Dmitry Yakubovich

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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 $L^2(-\pi, \pi)$. 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...

Source: http://arxiv.org/abs/1212.6014v1

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Sep 18, 2013
09/13

by
Anton Baranov; Alexander Borichev; Victor Havin

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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...

Source: http://arxiv.org/abs/math/0605052v1

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48

Sep 19, 2013
09/13

by
Alexander Borichev; Yuri Tomilov

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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.

Source: http://arxiv.org/abs/0910.0859v1

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Jun 28, 2018
06/18

by
Alexander Borichev; Andreas Hartmann; Karim Kellay; Xavier Massaneda

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We study multiple sampling, interpolation and uniqueness for the classical Fock space in the case of unbounded mul-tiplicities.

Topics: Mathematics, Complex Variables

Source: http://arxiv.org/abs/1508.03003

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38

Sep 24, 2013
09/13

by
Anton Baranov; Yurii Belov; Alexander Borichev

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We solve the spectral synthesis problem for exponential systems on an interval. Namely, we prove that any complete and minimal system of exponentials $\{e^{i\lambda_n t}\}$ 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 $\sum_n (f,\tilde e_n) e^{i\lambda_n t}$, where $\{\tilde e_n\}$ is the system biorthogonal to...

Source: http://arxiv.org/abs/1112.5551v2

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16

Jun 28, 2018
06/18

by
Alexander Borichev; Andreas Hartmann; Karim Kellay; Xavier Massaneda

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We study multiple sampling, interpolation and uniqueness for the classical Fock spaces in the case of unbounded multiplicities. We show that there are no sequences which are simultaneously sampling and interpolating when the multiplicities tend to infinity.

Topics: Mathematics, Complex Variables

Source: http://arxiv.org/abs/1508.03932

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47

Jul 20, 2013
07/13

by
Alexander Borichev

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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...

Source: http://arxiv.org/abs/1203.4731v1

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Jul 20, 2013
07/13

by
Alexander Borichev; Omar El-Fallah; Abdelouahab Hanine

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We use the so called resolvent transform method to study the cyclicity of the one point mass singular inner function in weighted Bergman type spaces.

Source: http://arxiv.org/abs/1304.0994v1

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Jun 28, 2018
06/18

by
Alexander Borichev; Artur Nicolau; Pascal J. Thomas

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Following Gorkin, Mortini, and Nikolski, we say that an inner function $I$ in $H^\infty$ of the unit disc has the WEP property if its modulus at a point $z$ is bounded from below by a function of the distance from $z$ to the zero set of $I$. This is equivalent to a number of properties, and we establish some consequences of this for $H^\infty/IH^\infty$. The bulk of the paper is devoted to "wepable" functions, i.e. those inner functions which can be made WEP after multiplication by a...

Topics: Functional Analysis, Mathematics, Complex Variables

Source: http://arxiv.org/abs/1508.01336

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47

Sep 19, 2013
09/13

by
Alexander Borichev

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We construct area-preserving real analytic diffeomorphisms of the torus with unbounded growth sequences of arbitrarily slow growth.

Source: http://arxiv.org/abs/math/0304199v1

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Sep 19, 2013
09/13

by
Alexander Borichev

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We obtain several results on the distortion asymptotics for the iterations of diffeomorphisms of the interval extending the recent work of Polterovich and Sodin.

Source: http://arxiv.org/abs/math/0304440v1

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38

Sep 19, 2013
09/13

by
Alexander Borichev; Artur Nicolau; Pascal J. Thomas

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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...

Source: http://arxiv.org/abs/math/0304482v2

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115

Sep 17, 2013
09/13

by
Alexander Borichev; Mikhail Sodin

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We extend two theorems of Krein concerning entire functions of Cartwright class, and give applications for the Bernstein weighted approximation problem.

Source: http://arxiv.org/abs/math/0007008v1

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Sep 22, 2013
09/13

by
Alexander Borichev; Don Hadwin; Hassan Yousefi

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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...

Source: http://arxiv.org/abs/1001.1018v2

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Sep 23, 2013
09/13

by
Alexander Borichev; Prabhu Janakiraman; Alexander Volberg

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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...

Source: http://arxiv.org/abs/1012.0943v3