Phone +34 93 586 8522
E-mail stikhonov@crm.cat
Position CRM Principal Investigator
Funding ICREA
Group Harmonic Analysis and Approximation Theory
Tikhonov, Sergey
Biosketch

About me

PhD (2003, Moscow State University) and ICREA researcher at CRM since October 2008

I was born in Russia in 1976. I graduated from the Lomonosov Moscow State University in 1999 and obtained my PhD in Mathematics from MSU in 2003.

2004-2006: Marie Curie Fellow at the Centre de Recerca Matemàtica, Barcelona.

2006-2008: Post-doctoral Fellow at the Scuola Normale Superiore, Pisa.

2008-September 2012: ICREA Researcher, CRM

From September 2012, I am an ICREA Research Professor at the Centre de Recerca Matemàtica.

2009: ISAAC Award. 2012: Humboldt Research Fellowship for Experienced Researchers.

 

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Interests

My research mainly deals with Fourier Analysis and Approximation Theory. Particularly, I study the relationship between "smoothness" of a function and a possibility to approximate or to represent this function by a sum of "simple" functions. The case when "simple" functions have wave structure is of special interest.

Currently my main interests include:

Harmonic Analysis: Fourier series/transforms; Function spaces; Embedding theorems; (Singular) Integral operators; Weights

Approximation theory: Polynomial approximation, Polynomial inequalities and applications, Orthogonal polynomials, Moduli of smoothness, K-functionals

Real Analysis: General monotone functions and sequences, Hardy-type inequalities

More specifically, I have been recently studying the following topics:

Weighted Fourier inequalities (classical Fourier transform): The main problem is to prove (Lp,Lq) weighted norm estimates for the Fourier transform. We study an extension of Pitt type and two-sided Boas type inequalities.

Weighted Fourier inequalities (general transforms): We investigate (Lp,Lq) weighted norm inequalities for the general integral transforms (Hankel, Mehler-Fock, etc ). Pitt and Hardy-Littlewood type estimates are studied.

Moduli of smoothness, Embedding theorems: Recently, we have found sharp interrelation between moduli of smoothness of different orders in various (Lp,Lq) metrics. Two cases p=q and p<q correspond to sharp Marchaud and Ulyanov inequalities, respectively. Both, in turn, are linked to sharp Jackson and inverse inequalities in approximation theory and to embedding theorems for Sobolev and Besov spaces. It turns out that for limiting cases when p=1 or/and q=infty, to investigate such interrelation requires new polynomial inequalities.

General monotone functions and Fourier transforms: We introduced the general monotone functions and sequences that have a number of applications in Fourier analysis and approximation theory. In particular, we apply some properties of general monotonicity to study Wiener type localization problems for Fourier series, coefficients criteria for Fourier series, integrability problems for Fourier transforms, equivalence of certain structural and constructive concepts in approximation theory.

Convolution operators in Lorentz spaces: We investigate norm convolution inequalities in Lebesgue and Lorentz spaces. First, we improve the well-known O’Neil estimate of norm of the convolution operator and obtain a corresponding estimate of this norm from below. Second, we prove O’Neil-type inequality for the Lorentz spaces in the limiting case (note that the first limiting case was studied by H. Brézis and S. Wainger). We prove similar inequalities in the weighted Lorentz spaces. We also establish norm inequalities for the potential operator in the weighted Lebesgue spaces.

Integral operators: We obtain necessary and sufficient conditions for the integral operators to be of strong or weak-type. For this purpose, we introduce new function spaces which we call the Net Spaces and which are natural generalizations of the Lorentz spaces. We obtain norm inequalities for the convolution operator in weighted Lebesgue spaces via the net spaces norm of the kernel.