Quadrature Formulas with High Order Approximation

In the article the method of creation the quadrature formulas with high order approximation for a wide class of the areas is given. This method is based on approach of smooth function on the plane by the semilocal smoothing spline or S-spline. Semilocal smoothing splines are initiated by D. A. Silaev. Earlier the splines of the third and fifth degree are considered and applied. This work is devoted to use of S-splines of higher degrees. Steady $S$-splines of a class of $C^0$ (only continuous), consisting of polynoms of high degree of $n$ ($n=9, 10$) makes it possible to receive quadrature formulas of the 10th and 11th orders of approximation. It is supposed that integrand function belongs to $C^p$ class (to $p=10, 11$) in a bigger area, than initial area on which integration is conducted. It is also supposed that the border of area is set parametrically that helps to consider area border with a fine precision. Similar approach is possible for the construction of cubature formulas.