Exact orders of computational (numerical) diameters in problems of reconstructing functions and sampling solutions of the Klein-Gordon equation from Fourier coefficients

Abstract

The study of real-world phenomena by mathematical methods is performed by the following scheme: on the basis of observations and experiments, a mathematical model is constructed; this model is investigated by mathematical means, and certain conclusions in the framework of the model are made, which are then compared with real facts (see, e.g., [1–6] and the references therein). In such a study, it should be taken into account that a computer makes it possible to store and process large finite arrays of numbers and performs four arithmetic operations on them. As a rule, the evaluation of functionals, which are the main carriers of numerical information, cannot be mathematically exact; at the same time, redundant computational precision in the implementation of an algorithm unjustifiedly increases the amount of memory and the number of arithmetic operations, because it does not improve the designed-in accuracy of the algorithm. A mathematical equivalent of these propositions is the notion of computational (numerical) diameter suggested in [7–13] (see also [14–24] and the references therein) in 1996–2003. It is aimed at determining the best computing facilities in a given class in the conditions of corrupted initial data. In the existing literature, there are at least three versions of the general setting of the problem of optimal reconstruction from inaccurate information. A common feature of these three settings is the intermediate problem of reconstructing an operator Tf( · ) by a computing aggregate