WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 13, 2014
On Construction Third Order Approximation Using Values of Integrals
Authors: ,
Abstract: Sometimes results of experiments contain values of integrals of a function over sub-intervals however one needs to construct a continuous approximation of the function on the interval. This problem may be solved by the polynomial or by the trigonometrical integro-differential splines that are constructed in this paper. The integro- differential splines are useful for construction the approximation of the function on each sub-interval separately. We use the values of the integrals of the function over sub-intervals and construct a continuous approximation of the function on the interval by the polynomial integro-differential spline, and by the trigonometric integro- differential spline in two steps. First we obtain discontinuous third order approximation of the function in form of the polynomial integro-differential spline, or the trigonometrical integro-differential spline. Approximation on sub-interval uses only the values of the integrals of the function to be approximated over three intervals and basic functions which we obtain here. Then we construct continuous approximation of the function on the interval by solving the system of equations. After that we compare the properties of solutions in form of the polynomial and in form of the trigonometric integro-differential splines. One can see that for trigonometric function sometimes the trigonometric integro-differential spline gives better approximation then the polynomial integro-differential spline. Finally we construct approximation of the function in case we use quadrature formula of the third order instead of the value of the integral over sub-interval.
Search Articles
Keywords: interpolation, splines, polynomial splines, trigonometric splines, polynomial integro-differential splines, trigonometric integro-differential splines, approximation, error of approximation
Pages: 676-683
WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 13, 2014, Art. #66