On approximation and convergence of one-dimensional parabolic integrodifferential polynomials and splines

The methods for approximation of functions with one-dimensional (1D) integrodifferential polynomials of the 2nd degree and derived conservative parabolic integrodifferential splines are considered. In majority of applied computational tasks, accuracy of source data does not exceed precision of approximation by parabolic polynomials and splines. The nodes of conventional parabolic splines, based on differential matching conditions with approximated function (further named as differential splines), are shifted relatively to interpolation nodes in order to provide stability of approximation process. The shift between spline and approximation nodes complicates computational algorithms drastically. Additionally, traditional differential splines are not conservative, i. e., they do not maintain integral characteristics of approximated functions. The novel integrodifferential parabolic splines that use integral deviation as criteria for matching a spline with a source function are presented. These splines are stable if spline nodes coincide with nodes of approximated functions and conservative with respect to sustaining area under curves. The theorems on approximation of mathematical functions with 1D integrodifferential parabolic polynomials and convergence of parabolic integrodifferential splines are proved. It is suggested to apply the proposed integrodifferential splines for development of mathematical data processing models for large area spread information systems.