ON ONE PROBLEM OF RECOVERING FUNCTIONS WHICH HAVE NO MORE THAN m INTERNAL EXTREMUMS - Nechyporenko, N. O.; KorotunovaО. V. (Computer Science and Applied Mathematics, 2023)
Restoring the function according to its experimental values, which are set at the nodes of a regular grid, is a classic problem of approximation theory. A large number of publications are devoted to the problem of constructing interpolation splines. The scope of such splines is limited to tables containing exact values of the function being interpolated. However, as a result of experiments, as a rule, approximate values are obtained, which may not correspond to the available a priori information about certain properties of the function. At the same time, in practice, there is often a need for the restored function to preserve certain properties that can be obtained from a priori ideas about the course of certain physical, economic, social processes or phenomena described by the sought functions. In such cases, standard spline approximation methods do not always provide a satisfactory solution to the function recovery problem. So, for example, preservation of monotonicity and convexity for interpolation splines can be achieved only with additional, fairly strict restrictions on the initial data and grid nodes. Tabular values must correspond to the geometric properties of the function. In the experiment, as a rule, “noisy” values of the function are registered, which most often do not correspond to the available a priori information. This means that in many cases, smoothing is a necessary step in the information processing process. Thus, the construction of optimal algorithms for the restoration of grid functions, which take into account a priori information about the geometry of the restored function, is an urgent task. The purpose of this work is to build algorithms for restoring functions that are given by their approximate values at the nodes of an arbitrary fixed grid and have a given number of extrema in the domain of definition. Restorative functions are built on the basis of the method of quasi-solutions. Step-bystep algorithms for constructing restorative functions for the specified class of functions are presented. These algorithms allow not only to preserve the geometric properties of the restored function, but also, as evidenced by the results of numerical experiments, to achieve sufficient restoration accuracy. If the specified accuracy of the input data and the corresponding class of functions is limited, then the given algorithms are optimal in order of accuracy with an order constant not exceeding 2.