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58

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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51

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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42

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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59

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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43

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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45

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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43

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

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50

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