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Vol 212, No 6 (2021)
On uniqueness of probability solutions of the Fokker-Planck-Kolmogorov equation
Abstract
The paper gives a solution to the long-standing problem of uniqueness for probability solutions to the Cauchy problem for the Fokker-Planck-Kolmogorov equation with an unbounded drift coefficient and unit diffusion coefficient. It is proved that in the one-dimensional case uniqueness holds and in all other dimensions it fails. The case of nonconstant diffusion coefficients is also investigated. Bibliography: 70 titles.
Matematicheskii Sbornik. 2021;212(6):3-42
3-42
Characterization of solutions of strong-weak convex programming problems
Abstract
Finite-dimensional problems of minimizing a strongly or weakly convex function on a strongly or weakly convex set are considered. Necessary and sufficient conditions for solutions of such problems are presented, which are based on estimates for the behaviour of the objective function on the feasible set taking account of the parameters of strong or weak convexity, as well as certain local features of the set and the function. The mathematical programming problem is considered separately for strongly and weakly convex functions. In addition, necessary conditions for a global and a local solution with differentiable objective function are found, in which a ‘strong’ stationarity condition is assumed to hold. Bibliography: 33 titles.
Matematicheskii Sbornik. 2021;212(6):43-72
43-72
Multivariate Haar systems in Besov function spaces
Abstract
We determine all cases for which the $d$-dimensional Haar wavelet system $H^d$ on the unit cube $I^d$ is a conditional or unconditional Schauder basis in the classical isotropic Besov function spaces ${B}_{p,q,1}^s(I^d)$, $0< p,q< \infty$, $0\le s < 1/p$, defined in terms of first-order $L_p$-moduli of smoothness. We obtain similar results for the tensor-product Haar system $\widetilde{H}^d$, and characterize the parameter range for which the dual of ${B}_{p,q,1}^s(I^d)$ is trivial for $0< p< 1$.
Bibliography: 31 titles.
Matematicheskii Sbornik. 2021;212(6):73-108
73-108
Recovery of integrable functions and trigonometric series
Abstract
Classes $\Gamma$ of $L_1$-functions with fixed rate of decrease of their Fourier coefficients are considered. For each class $\Gamma$, it is shown that there exists a (recovery) set $G$ with arbitrarily small measure such that any function in $\Gamma$ can be recovered from its values on $G$. A formula for evaluation of the Fourier coefficients of such a function from its values on $G$ is given. In addition, it is shown that, for any $L_1$-function, a function-specific recovery set can be found. The problem of recovery of general trigonometric series from the Zygmund classes which converge to summable functions on such sets $G$ is also solved. Bibliography: 10 titles.
Matematicheskii Sbornik. 2021;212(6):109-125
109-125
On defining functions and cores for unbounded domains. III
Abstract
We extend the authors' results on existence of global defining functions to a number of different settings. In particular, we relax the assumption on strict pseudoconvexity of the domain to strict $q$-pseudoconvexity and we consider more general situations, where the ambient space is an almost complex manifold or a complex space. We also investigate to what extent the assumption on smoothness of the boundary of the domains under consideration is necessary in our results. Bibliography: 27 titles.
Matematicheskii Sbornik. 2021;212(6):126-156
126-156