Integro-Differential Splines and Quadratic Formulae

Abstract: This work is devoted to the further investigation of splines of the fifth order approximation. Here we present some new formulae which are useful for the approximation of functions with one or two variables. For each grid interval (or elementary rectangular) we construct the approximation separately. Here we construct the basic one-dimensional polynomial splines of the fifth order approximation when the values of the function and the values of its first derivative are known in each point of interpolation. Sometimes it is important that the integrals of the function over the intervals are equal to the integrals of the approximation of the function over the intervals. In that case the approximation has some physical parallel. For this aim we use quadratic formulae here with the sixth order of approximation instead of the value of integral. The one-dimensional case can be extended to multiple dimensions through the use of tensor product spline constructs. Numerical examples are represented.