Approximations of Nonlinear Integral Functionals of Entropy Type
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proximations of Nonlinear Integral Functionals of Entropy Type V. I. Bogachev a,b Received February 20, 2020; revised February 20, 2020; accepted April 6, 2020
To Valery Vasil’evich Kozlov on the occasion of his 70th birthday Abstract—We obtain generalizations and strengthenings of the results of V. V. Kozlov and D. V. Treschev on approximations of nonlinear integral functionals of entropy type on measure spaces. DOI: 10.1134/S0081543820050016
1. INTRODUCTION The paper is devoted to generalizations and strengthenings of several results of V. V. Kozlov and D. V. Treschev obtained in [10–12] and related to the study of approximations of nonlinear integral functionals of entropy type. This study was in turn motivated by the investigation of the evolution of dynamical systems. The complex of problems studied by these authors also included the analysis of nonuniform averagings in ergodic theorems generated by linear integral operators, which is the subject of my separate survey [3]. The cited papers have attracted attention of other researchers and have been further developed (see [4, 16]). The nonlinear integral functionals under study have the form g → F (x, g(x)) μ(dx), X
where F is a measurable function that is convex in the second argument (or is dominated by a convex function) and the approximating functionals have the form F (x, Tj g(x)) μ(dx) X
with some linear operators Tj (of the type of conditional expectations) or even the more complicated form F (x, Tj gn (x)) μ(dx) X
with linear approximations Tj gn for functions gn converging to g. A specific feature of this problem in the second version is that the functions gn converge only weakly in L1 (μ); however, the operators Tj that have the form of conditional expectations help to improve the convergence of the compositions a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991
Russia. b National Research University Higher School of Economics, ul. Myasnitskaya 20, Moscow, 101000 Russia.
E-mail address: [email protected]
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F (x, Tj gn (x)) to the strong one, that is, the norm convergence. We also consider the case of an even weaker convergence of functions that corresponds to weak convergence of measures. This method of recovering the value of the functional on the function g by using functions gn that converge to g is explained by the fact that the straightforward approximations by the integrals of F (x, gn (x)) may fail to converge or may converge to a wrong value because of a rather weak convergence of gn . For example, the functions sin(2πnx) converge weakly to zero in L1 [0, 1], but their squares converge weakly to 1/2, which gives a wrong answer for F (x, y) = y 2 . In their setting, the problems discussed here differ from rather close problems related to the calculus of variations (see [6–9]). 2. NOTATION, TERMINOLOGY, AND AUXILIARY RESULTS Throughout the paper we employ standard concepts and facts (see [1, 2, 5]) related to the Lebesgue integral and the space L1 (μ) with re
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