Adaptive approximation (or interpolation) takes into account local variations in the behavior of the given function, adjusts the approximant depending on it, and hence yields the smaller error of approximation. The question of constructing optimal approximating spline for each function proved to be very hard. In fact, no polynomial time algorithm of adaptive spline approximation can be designed and no exact formula for the optimal error of approximation can be given. Therefore, the next natural question would be to study the asymptotic behavior of the error and construct asymptotically optimal sequences of partitions. In this paper we provide sharp asymptotic estimates for the error of interpolation by splines on block partitions in R d . We consider various projection operators to define the interpolant and provide the analysis of the exact constant in the asymptotics as well as its explicit form in certain cases.

In this paper, we derive a generalized multiplicative Hardy-LittlewoodPolya type inequality, as well as several related additive inequalities, for functions of operators in Hilbert spaces. In addition, we find the modulus of continuity of a function of an operator on a class of elements defined with the help of another function of the operator. We then apply the results to solve the following problems: (i) the problem of approximating a function of an unbounded self-adjoint operator by bounded operators, (ii) the problem of best approximation of a certain class of elements from a Hilbert space by another class, and (iii) the problem of optimal recovery of an operator on a class of elements given with an error.Keywords Inequalities of Hardy-Littlewood-Polya type · functions of operators · modulus of continuity · best approximation of unbounded operators · optimal recovery of operators Mathematics Subject Classification (2000) MSC 26D10 · MSC 47A63 · MSC 41A17 · MSC 47A58

No abstract

We prove sharp Ostorowski type inequality for multivariate Sobolev classes and apply it to the problem of optimal recovery of integrals.Mathematics subject classification (2010): 26D10, 41A17, 41A44, 41A55.

Abstract. In this paper we present the solution to the problem of recovering rather arbitrary integral operator based on incomplete information with error. We apply the main result to obtain optimal methods of recovery and compute the optimal error for the solutions to certain integral equations as well as boundary and initial value problems for various PDE's.Key words. optimal recovery, approximation, information with error, integral operators, integral equations, initial and boundary value problems AMS subject classifications. 41A35, 45P05, 35G151. Introduction. Solutions to boundary (or initial) value problems for various partial differential equations require knowledge of a boundary (or initial) function. However, often time, those functions are not fully known and only partial information about them can be measured, e.g. values at some finite set of points, average values over small measurement intervals, values of N first consecutive Fourier coefficients, etc. Thus, it is very important to find an approximate solution based on available information on the boundary (or initial) function. Furthermore, it is also natural and important to develop methods that provide an optimal (in some sense) approximation to the true solution. These research questions have been explored under the theory of optimal recovery of functions and operators, which is an area of Approximation Theory that started to develop in 1970s. More information on the development of the area can be found, for instance, in [20,28,21,12,17,24,25,11].As for specific applications to recovering solutions of boundary and initial value problems, Magaril-Ill'yaev, Osipenko, and co-authors (see, for instance, [16,22,18]) have considered the problem of optimal L 2 -approximation of the solution to the Dirichlet problem for Laplace's and Possion's equations in simple domains (disk, ball, annulus) based on the first N consecutive Fourier coefficients of the boundary function (possibly given with an error). In order to solve this problem they have used methods of Harmonic Analysis and general results from Optimization Theory.In this paper we address related questions of optimal approximation of the solution to several types of integral equations, boundary and initial value problems for PDE's. We begin by solving a more general problem of recovering a rather arbitrary integral operator and sum of operators. We then present the optimal method of recovery as well as the optimal error. Next, we apply this general result to recover solutions to various boundary and initial value problems. Moreover, we present optimal methods of recovery of the solution to boundary-value problems based on this incomplete information with error. Naturally, the solution to the problem when information with error is used will also lead to the solution to the problem with exact information. In this paper we focus on considering the Volterra's and Fredholm's linear integral equations as well as boundary value problems for wave, heat, and Poisson's equa-