Probabilistic Methods in Analysis

Scientific results:

• The leading scientist and the laboratory team have carried out work on the study of the Coulomb gas model near the spectral boundary and related problems of the theory of orthogonal polynomials. Exact asymptotics of orthogonal polynomials corresponding to weighted spaces in a bounded domain of the complex plane has been obtained.  The nonlinear Riemann-Hilbert problem is solved using a generalization of the Its-Takhtajan approach. A research is conducted of the diagonal behaviour for the Bergman kernel corresponding to the correlation kernels for random matrices.
• Upper estimates of the probability of occurrence of large clusters of particles have been proved for determinant point processes in the real and complex cases. It has been shown that the behavior of determinant point processes differs greatly from models in which particles are independent; namely, in the determinant case, large clusters are much less probable. Thus, the Coulomb gas exhibits much more regular behavior than, for example, the Poisson process or multidimensional Brownian motion. An explicit description of the Palm measures of determinant processes with a confluent hypergeometric kernel, naturally arising in the problem of harmonic analysis on an infinite-dimensional unitary group, has been obtained.
• A description of complete interpolation sequences for the shift-invariant space generated by the shift of the Gaussian function is obtained. The criterion found has a simple geometric form and is expressed in terms of average deviations from an integer lattice. It is also shown that any sampling sequence contains a complete interpolation sequence, and any interpolation sequence can be complemented to a complete interpolation sequence, which allows us to prove in a new way the well-known results of Groechenig, Romero, and Stokler (2018) on the description of sampling and interpolation sequences in terms of upper and lower densities.
• A problem of B.S. Kashin on the behavior of the Schatten-von Neumann norms of the projection operator in the space of matrices of a given dimension onto upper-diagonal triangular matrices was solved, asymptotically sharp estimates are obtained for the Schatten-von Neumann type norm of the projection operator.
• The Birman-Krein-Vishik theory of extensions of a non-negative symmetric operator is substantially extended. It is shown that the resolvent of the reduced Krein extension of the operator and the reduced resolvent of the operator itself are compact only simultaneously, and the eigenvalues ​​of these operators have the same asymptotics. The class of symmetric Jacobi matrices that do not satisfy the Carleman condition was investigated. New conditions for the self-adjointness of such matrices were found, as well as conditions for the discreteness of their spectra and the maximality of the deficiency indices. These results are applied to prove new conditions of self-adjointness, discreteness, and maximality of the deficiency indices of the Schroedinger and Dirac operators with point interactions.
• New estimates of integral means of derivatives of rational and n-valented functions in domains with fractal boundaries are obtained. E.P. Dolzhenko's inequalities are generalized to the case of n-valent functions in domains with rectifiable boundaries. For domains whose boundary has a dimension greater than one, power estimates are found depending on the order of valency, in which the exponent depends on the Minkowski dimension of the domain.
• A new method for constructing representing systems of reproducing kernels for a wide class of spaces of analytic functions is proposed. As an application, representing systems of Cauchy kernels are constructed in Hardy spaces in the disk, in the disk-algebra, and also for a wide class of weighted Hardy spaces (including, in particular, Dirichlet, Bergman, and Hardy-Sobolev spaces). A recursive procedure for finding coefficients and an estimate of the convergence rate are presented.

Implementation of research results:

We have obtained results of the research of the uncertainty principle in problems color correction performed for an industrial partner: we researched transformations between the RGB and XYZ color spaces, proposed several types of such transformation as well as variations of the cost function and an algorithm for finding its global minimum and tested these developments over an array of real-life data. This can be used in finding algorithms of linear transformation with set limitations, recalculating color tables with differentiation of various color parameters.

Education and personnel occupational retraining:

The following scientific conferences were organized: «Complex and Harmonic Analysis and Its Applications» (Saint Petersburg, November 23-26, 2021), «Probabilistic Techniques in Analysis: Spaces of Holomorphic functions» (Sochi, Sirius, December 6-10, 2021), «Probabilistic Techniques in Analysis: Reproducing kernel Hilbert spaces» (Sochi, Sirius,  October 20-25, 2022)​, «Days of Analysis in Sirius» (Sochi, Sirius, October 16-20​, 2023​), «Discrete and Continuous Signals: Analysis, Information and Applications» (Saint Petersburg, December 11-16​, 2023​).

In the framework of the project, the following were defended: dissertation for the academic title of Doctor of Physical and Mathematical Sciences – 1, dissertation for the academic title of Candidate of Physical and Mathematical Sciences – 1.

Cooperation:

• Sirius Mathematical Center (Russia)
• The Euler International Mathematical Institute, Saint Petersburg (Russia)
• Almazov National Medical Research Centre (Russia)