HARMONICS AND SPLINES AS OPTIMAL TOOLS FOR APPROXIMATION AND RECOVERY

The basic notions of mathematics (such as number, set, function) usual ly carry an infinite amount of information. To make practical use of t his information, we have to 'finitize' it (that is, code it using fini tely many of the simplest units of information) in such a way that we can later recover the initial function or set with given accuracy. In this connection there naturally arises the problem of an optimal codin g, approximation, and recovery of such objects. In the majority of cas es, the problem of best methods for the approximation and recovery of functions leads to the consideration of two types of spaces: the space of harmonics and/or the space of splines. The purpose of this article is to describe classes of problems in which harmonics and splines are tools for the optimal approximation or recovery of functions (in the exact or in an asymptotic sense). We have tried to follow the route fr om basic statements to sufficiently wide generalizations of these. We have to say that not everything has received its final form yet. Takin g this and the large amount of material into account we will give cert ain descriptions without sufficient detail, postponing to future publi cations the treatment of a number of questions considered here. The ba sis of this article is a lecture given by the author during the confer ence celebrating the 90th anniversary of the birth of A.: N. Kolmogoro v (St. Petersburg, June 1993).