ON UNIFORM RESTORATION OF FUNCTIONS WHICH HAVE NO MORE THAN TWO POINTS OF INFLECTION - Nechyporenko, N. А.; Korotunova, O. V. (Computer Science and Applied Mathematics, 2016)
The problem of tabular data approximation is one of the major problems arising in the processing of the experimental results. In many cases, the experimenter has additional information about the geometric properties of the restored function, which are advisable to take into account and save. Many works are dedicated to the constructing interpolation isogeometric splines problem. However, the application area of these splines is limited to tables containing the exact values of the interpolated function. Generally, in practice there are approximate values that do not match the available priori information about the geometric properties of the functions. Therefore, the problem of restoration functions algorithms constructed by theirs approximate gridded data and a priori information about theirs geometric properties is actual. This work considers the problem of restoring a continuous function from a class of functions defined by theirs approximate values in arbitrary fixed grid nodes. As a set of functions, were considered the following sets: concave and convex monotone; having a domain not more than two points of inflection and monotone function with the same property. Restoring function is based on the quasi-solutions method. The step-by-step algorithms for restoring the function of each of the classes of functions are provided. The numerical results show that these algorithms can achieve high accuracy of the recovery. If the uncertainty of the input data at the grid is known and the corresponding class of functions is limited, it is possible to construct upper and lower bounds of the corresponding class of functions. Then, as is known, for optimum accuracy of approximation in the class will be a Chebyshev center of a class equal to half the sum of the upper and lower boundaries of the class. However, in this case the desired geometric properties of functions will not be performed, which is unacceptable. Quoted in the work function approximation algorithms retain its geometric properties and are optimal in order of accuracy, with the constant orders not exceeding 2.