Seminar in complex and harmonic analysis

Karol Dziedzul, Gdansk University of Technology

Smooth orthogonal projections and Parseval wavelet frames on Riemannian manifold

П.А Бородин (МГУ)

Приближение наипростейшими дробями

Будет сделан обзор (в основном качественных) результатов по приближению наипростейшими дробями  (логарифмическими производными комплексных многочленов) в различных функциональных пространствах и связанных с ними результатов из теории плотности полугруппы в банаховых пространствах.

Nikos Chalmoukis, University of Bologna

Onto Interpolation for the Dirichlet space and for H_1(D)

We will discuss a characterization of onto interpolating sequences with finite associated measure for the Dirichlet space in terms of capacity of some condensers. The same condition in fact characterizes all onto interpolating sequences for H_1(D) even if the associated measure is infinite.

Alexander Borichev, University Aix-Marseille

Zeros of entire functions with random and pseudo-random Taylor coefficients 

We study the zero distribution for entire functions with random and pseudo-random Taylor coefficients and related problems on stationary sequences and polynomial approximation.

Nikos Chalmoukis, University of Bologna

Totally null sets and capacity in Dirichlet type spaces 

In the context of Dirichlet type spaces on the unit ball of Cd, also known as Hardy-Sobolev or Besov-Sobolev spaces, we compare two notions of smallness for compact subsets of the unit sphere. We show that the functional analytic notion of being totally null agrees with the potential theoretic notion of having capacity zero. In particular, this applies to the classical Dirichlet space on the unit disc and logarithmic capacity. In combination with a peak interpolation result of Davidson and the second named author, we obtain strengthenings of boundary interpolation theorems of Peller and Khrushchëv and of Cohn and Verbitsky.

Artur Nikolau, University of Barselona

A Central Limit Theorem for iterates of Inner Functions 

A Central Limit Theorem for linear combinations of iterates of an inner function is proved. The main technical tool is Aleksandrov Desintegration Theorem for Aleksandrov-Clark measures.

Joint work with Odi Soler

N. Arcozzi, University of Bologna

The Drury-Arveson space on the Siegel domain

 The Drury-Arveson space in the unit complex ball and its role in operator theory has been object of intensive investigation for the last twenty years. We suggest a definition of a “flat” version of the space on the Siegel domain and consider some of its connections with the Heisenberg group and its representations, and sub-Riemannian geometry. Most of the work was in collaboration with Alessandro Monguzzi, Marco Peloso, and Maura Salvatori.

P. Perstneva, St.Petersburg State University

Indicator functions with uniformly bounded Fourier sums and large gaps in the spectrum

 We will speak on the recent work, closely related to the well-known article of Olevskii and Nazarov from 2017. We will present the construction of an indicator function with “thin” Fourier-spectrum and bounded “Fourier sums” on the line (on an arbitrary locally-compact Abelian group of finite dimension), which can be viewed as a “violation” of the uncertainty principle. Moreover, such an indicator function can be obtained by small perturbation from any indicator function of a set of finite measure fixed in advance. Therefore the result can also be interpreted as a Men’shov-type correction theorem. Recall that the classical Men’shov theorem asserts that any continuous function on the circle can be corrected on a subset of an arbitrary small measure to get a function with uniformly convergent Fourier series.

We plan to discuss the notions of “thin” Fourier-spectrum and “Fourier sums” for locally compact Abelian groups and to state a certain weighted version of the result. Then we will sketch the proof of the result above for the line. We will also present another Men’shov-type result for essentially bounded functions with compact support on the line, which ensures a sharp estimate on the supremum of partial Fourier integrals.

Joint work with S.V. Kislyakov, https://arxiv.org/abs/2006.02561

A. Kulikov, Norwegian University of Science and Technology and SPbSU

Fourier interpolation and time-frequency localization

In their recent breakthrough paper D. Radchenko and M. Viazovska proved that every Schwartz function can be recovered from the values of it and its Fourier transform at by means of an interpolation formula

If we consider the corresponding interpolation sets and their counting functions , we can easily see that , which perfectly matches famous Slepian’s Theorem.

Recently, we showed (A. Kulikov, arXiv:2005.12836) that a similar bound

holds for all such Fourier interpolation formulas. The proof is based on the properties of the so-called prolate spheroidal wave functions, which are interesting in their own right.

A. Bondarenko, Norwegian University of Science and Technology

Fourier interpolation with zeros of zeta and -functions

We construct a large family of Fourier interpolation bases for functions analytic in a strip symmetric about the real line. Interesting examples involve the nontrivial zeros of the Riemann zeta function and other -functions. We establish a duality principle for Fourier interpolation bases in terms of certain kernels of general Dirichlet series with variable coefficients. Such kernels admit meromorphic continuation, with poles at a sequence dual to the sequence of frequencies of the Dirichlet series, and they satisfy a functional equation. Our construction of concrete bases relies on a strengthening of Knopp’s abundance principle for Dirichlet series with functional equations and a careful analysis of the associated Dirichlet series kernel, with coefficients arising from certain modular integrals for the theta group.

A. Baranov, St.Petersburg State University

Аппроксимация функциями из диск-алгебры в пространствах аналитических функций, инвариантных относительно обратного сдвига

В докладе планируется рассказать об одной недавней работе А. Алемана и Б. Мальмана о плотности функций из диск-алгебры в гильбертовых пространствах функций, аналитических в круге. Показано, что такая плотность имеет место для класса пространств, в которых оператор обратного сдвига является сжатием. Этот класс включает в себя модельные пространства и пространства де Бранжа-Ровняка. 

J. Ortega Cerda, University of Barcelona

An enhanced uncertainty principle

We improve on some recent results of Sagiv and Steinerberger that quantify the following uncertainty principle: for a function f with mean zero, then either the size of the zero set of the function or the cost of transporting the mass of the positive part of f to its negative part must be big. We also provide a sharp upper estimate of the transport cost of the positive part of an eigenfunction of the Laplacian. This proves a conjecture of Steinerberger and provides a lower bound of the size of a nodal set of the eigenfunction. Finally, we use a similar technique to provide a measure of how well the points in a design in a manifold are equidistributed.

This is joint work with Tom Carroll and Xavier Massaneda.

O. Yakir, Tel Aviv University

Recovering the lattice from its random perturbations

Given a d-dimensional Euclidean lattice, we consider the point process obtained by adding an independent Gaussian vector to each of the lattice points. In my talk, I will explain a simple procedure that recovers the underlying lattice from a single realization of this random point process (arXiv:2002.01508).

I. Bochkov, St.Petersburg State University

Fourier interpolation on the real line

Мы разберем работу М. Вязовской и Д. Радченко про интерполяцию в классе Шварца: “We use weakly holomorphic modular forms for the Hecke theta group to construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a function at any given point in terms of the values of the function and its Fourier transform on the set .”

K. Fedorovski, Moscow Bauman University

Знакомые и незнакомые свойства полианалитических функций

Полианалитические функции – это функции комплексного переменного, имеющие вид многочленов некоторой фиксированной степени от сопряженного переменного с голоморфными функциями в качестве коэффициентов. Класс полианалитических функций возник более 100 лет тому назад в связи с задачами плоской теории упругости. Полианалитические функции представляют значительный интерес для теории эллиптических дифференциальных уравнений в частных производных, так как многие свойства функций этого класса являются “модельными” для решений уравнений, не являющихся сильно-эллиптическими.  Возникают полианалитические функции и в многомерном комплексном анализе при изучении свойств функций двух комплексных переменных на неаналитических поверхностях.  В докладе будет приведено введение в теорию полианалитических функций и представлен ряд недавних результатов в этой теории.

P. Mozolyako, St.Petersburg State University

Discrete and continuous Carleson embeddings for Hilbert spaces of harmonic and analytic functions.

One of the possible ways to interpret the description of the trace measure-weight pair for the weighted Hardy inequality on a tree (and poly-tree) is the characterization of Carleson measures for the weighted Dirichlet spaces on the unit disc (poly-disc). In dimension one this connection was investigated by Arcozzi, Rochberg et al., in higher dimensions however the only known results concern the endpoints of the Hardy-Dirichlet scale, in particular the Hardy space case (in this context) is treated via a highly non-trivial result by Lacey-Ferguson on the symbols of Hankel operators in the bidisc. We aim to give a short overview of this discrete-continuous-analytic transition and discuss some arising questions.

Yu. Lyubarski, St.Petersburg State University

Системы экспонент в плоских выпуклых областях

Следуя работе Иосевича, Каца и Тао, мы докажем, что в пространстве суммируемых с квадратом функций в выпуклой области, граница которой имеет положительную кривизну, не существует ортогонального базиса из экспонент. Мы также обсудим связь этих вопросов с задачами интерполяции в пространствах целых функций двух переменных и теоремой Феффермана о мультипликаторе.

Yu. Belov, St.Petersburg State University

Задачи единственности для преобразования Фурье

Недавно М. Вязовская и Д. Радченко доказали, что если функция из класса Шварца и ее преобразование Фурье равны 0 в точках , то функция тождественно равна 0. Более того, это результат точен — ни одну точку нельзя выкинуть. Доказательство этой замечательной теоремы основано на теории модулярных форм и поэтому не может быть применено для возмущений исходного множества или даже намного более плотных множеств. Мы обсудим похожие теоремы единственности, полученные Дж. Рамосом и М. Суза, которые опираются на комплексный анализ. Множества единственности в этих теоремах намного более плотные, зато допускают возмущения.