Complex Analysis: theory of approximations by analytic and harmonic functions, best approximations;
applications of uniform and tangential approximations in various fields of complex analysis; investigation of problems of Weierstrass theory of analytic functions; Banach algebras of analytic functions; uniqueness problems of analytic and harmonic functions; value distribution theory of analytic and meromorphic functions; boundary value theory and boundary behavior of analytic, harmonic and subharmonic functions; integral transformations theory in complex domain; integral representations and classes of analytic and harmonic functions in multidimensional domains.
Real Analysis: trigonometric and general orthogonal series; bases in functional spaces; weighted functional spaces; differentiation of multidimensional integrals; representation and uniqueness for multiple Haar, Franklin, Walsh and trigonometric series; nonlinear approximation.
Probability Theory: integral and stochastic geometry; combinatorial integral geometry; point processes; sections of convex bodies by random planes and lines; measures generation by finite additive functionals; mathematical problems of statistical physics; limit theorems for random Gibbs processes and fields; statistics of stationary Gaussian processes.
Differential and Integral Equations: methods and algorithms for solution of equations; accelerating the convergence of decompositions by eigenfunctions of boundary problems and asymptotic estimates of the corresponding errors; computer realization of integral transforms and applications; parallelization of computations.
Mathematical Physics: methods of study and effective numerical-analytical solution of integral, integral-differential and other equations, arising in direct and inverse problems of radiative transfer, kinetic theory of gases, renewal stochastic processes, semi-Markov processes, filtration of stochastic processes, non-local interaction of waves; development of method of nonlinear factorization equations, Ambartzumian equation method; fixed point principles in the critical case.
Approximation theory: The solution of problems on possibility of uniform and tangential approximation by entire and meromorphic functions and functions holomorphic in a fixed domain; best and optimal uniform and tangential approximation by entire and meromorphic functions, applications in Complex Analysis. Uniform approximation by lacunary polynomials and entire functions. Regularity of bivariate Hermite interpolation schemes.
Real Analysis: Convergence of Trigonometric and general Fourier series. Representation and uniqueness for Haar, Franklin, Walsh and multiple trigonometric series. Construction of bases in Banach and weighted spaces. Differentiation of multidimensional integrals. Problems concerning densities of lacunar subsystems of orthogonal systems.
Complex Analysis: Integral Transforms, Integral representation and Classes of functions, boundary behaviour and uniqueness. Nevanlinna theory, Geometric function theory, efficient analytic continuation and singularities.
Probability theory and Mathematical Statistics: Integral and stochastic geometry: combinatorial integral geometry; point processes of intersections and stereological problems for geometric processes; Measure generation by functional in classical geometric spaces. Mathematical problems of statistical physics: limit theorems for weakly dependent random variables, Gibbs random processes and fields; asymptotic properties of spin and phase transitions in lattice systems. Statistics and prediction of stationary processes: Toeplitz type random functionals and forms; applications in the parametric and nonparametric estimation problems and testing statistical hypotheses.
Differential and Integral equations: Sufficient conditions of correctness of initial and mixed problems for the equations with multiple characteristics are established. Conditions of global solvability of some nonlinear partial differential equations are obtained. Explicit values of partial indices of factorization of matrix-functions of definite classes are found. Some classical problems of differential equations with infinite number of variables are investigated. Asymptotic values of L_2 errors for wavelet-like decompositions are investigated. Parallel processing algorithms for a number of investigated problems are developed.
Joint scientific investigations are carried out together with the following Scientific Centers: Steklov Mathematical Institute (Russia), Moscow University (Russia), Kazan University (Russia), University of Oldenburg (Germany), University of Trier (Germany), University of Freiberg (Germany), University of Le Mans (France), University of Paris-VI (France), University of Lyle (France), University of Kamerino (Italy), University of Madrid (Spain), Tsucuba University (Japan), Tample University (USA), Northeastern University (USA), Royal Institute of Technology (Sweden).
Institute has participated in the: International congresses of mathematicians; International and All-Union Symposiums and Conferences on real and complex analysis; International and All-Union Symposiums and Conferences on harmonic analysis and approximation theory; International Congresses on mathematical physics; International and All-Union Symposiums and Conferences on probability and statistics, on differential and integral equations.
Univariate and multivariate trigonometric and general orthogonal series. Bases in functional spaces, approximations.
Investigation of global properties of analitic functions by means of local data, applications.
Integral and stochastic geometry.
Differential and integral operators, applications.
Random fields and asymptotic problems of statistical phisics.
Some geometric problems of complex analysis.
Weighted spaces of regular functions associated with integro-differentiation, applications.
Investigation of linear and causal filters
Journal of Contemporary Mathematical Analysis
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