91
91

Jul 20, 2013
07/13

by
Alexander Borichev; Omar El-Fallah; Abdelouahab Hanine

texts

#
eye 91

#
favorite 0

#
comment 0

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

5
5.0

Jun 30, 2018
06/18

by
Alexander Borichev; Alon Nishry; Mikhail Sodin

texts

#
eye 5

#
favorite 0

#
comment 0

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

60
60

Jun 30, 2018
06/18

by
Charles Batty; Alexander Borichev; Yuri Tomilov

texts

#
eye 60

#
favorite 0

#
comment 0

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

59
59

Sep 20, 2013
09/13

by
Alexander Borichev; Prabhu Janakiraman; Alexander Volberg

texts

#
eye 59

#
favorite 0

#
comment 0

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

45
45

Sep 24, 2013
09/13

by
Yuri Bilu; Alexander Borichev

texts

#
eye 45

#
favorite 0

#
comment 0

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

43
43

Sep 23, 2013
09/13

by
Alexander Borichev; Prabhu Janakiraman; Alexander Volberg

texts

#
eye 43

#
favorite 0

#
comment 0

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

48
48

Sep 23, 2013
09/13

by
Alexander Borichev; Grigori Rozenblum

texts

#
eye 48

#
favorite 0

#
comment 0

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

50
50

Sep 22, 2013
09/13

by
Alexander Borichev; Mikhail Sodin

texts

#
eye 50

#
favorite 0

#
comment 0

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

44
44

Sep 18, 2013
09/13

by
Alexander Borichev; Haakan Hedenmalm

texts

#
eye 44

#
favorite 0

#
comment 0

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