Cubature and quadrature formulas of high order of approximation

This paper is concerned with the use of semilocal smoothing splines (or $S$-splines) for constructing cubature formulas. Such a spline is a piecewise-polynomial function, the first coefficients of each of the polynomials are determined by the smooth joint conditions, and the remaining ones, by the least-squares method. Previous studies were concerned with splines of degree $3$ and $5$. In the present paper, we consider $S$-splines of degree $n$ ($n=9,10$). Of special importance for calculation of integrals are the $S$-splines of class $C^0$ (the continuous ones). Such splines are employed in building quadrature and cubature formulas of high order of approximation for calculation of one-, two-, and three-dimensional integrals in a simply connected domain to $10$th and $11$th orders of approximation. The integrable function is assumed to lie in the class $C^{(n+1)}$ ($n=9,10$) in a somewhat larger domain than the original one (in which the integration takes place). It is also assumed that the boundary of the domain is given parametrically. This makes it possible to take into account, with high order of accuracy, the boundary of the domain. The corresponding convergence rates are estimates. A similar approach is also capable of building formulas for integration of smooth functions in multidimensional domains.