Exact inequalities and optimal recovery by inaccurate information
K. Yu. Osipenko
TL;DR
The paper develops an exact, multi-operator optimal recovery framework for recovering a weighted operator Λ_0 from noisy observations of a family {Λ_j} in multidimensional settings. By formulating the problem in terms of convex hulls Q and support functions S, it derives the exact recovery error E = e^{-S(α^0)} and characterizes optimal recovery maps as linear combinations with frequency- or space-domain constraints. This leads to exact Carlson-type inequalities with sharp constants, connecting optimal recovery to generalized Hardy–Littlewood–Polya-type bounds for multipliers and Weyl derivatives. The results unify recovery of differential operators, Fourier multiplier operators, and their associated exact inequalities, providing explicit constructions and extending classical inequalities to multi-parameter settings with exactness guarantees.
Abstract
The paper considers a multidimensional problem of optimal recovery of an operator whose action is represented by multiplying the original function by a weight function of a special type, based on inaccurately specified information about the values of operators of a similar type. An exact inequality for the norms of such operators is obtained. The problem under consideration is a generalization of the problem of optimal recovery of a derivative based on other inaccurately specified derivatives in the space $\mathbb R^d$ and the problem of an exact inequality, which is an analogue of the Hardy-Littlewood-Polya inequality.