Optimal cubature formulas for integration along a ball, which use information along concentric spheres
Sergiy Borodachov
Department of Mathematics
Georgia Institute of Technology
I plan to talk on optimal cubature formulas, which recover the integral of a function along a d-dimensional ball from its mean values along n concentric (d-1)-dimensional spheres inside the ball. We look for the best radii of the node spheres and the best weights. Optimality of the cubature formula is understood in the sense of minimal worst case error over certain classes of differentiable functions. In particular, we consider the class of functions, with the absolute value of the gradient bounded in L1-norm and the class of functions with bounded L1-norm of the Laplacian, which also vanish on the boundary of the ball. This problem is an extension of the classical Kolmogorov-Nikol'lkii problem about optimal quadratures.