Chebyshev-Type Quadrature on Multidimensional Domains Original Research Article

Quadrature formulas with equal coefficients for interval and circle are combined to obtain Chebyshev-type quadrature formulas (relative to ordinary area or volume measure) for "product domains." Upper bounds for the minimal number N = N(p) of nodes required for polynomial exactness to degree p readily follow. Lower bounds are obtained by projecting onto certain subsets of lower dimension and other means. The precise order of N(p) is determined for square, cube, cylindrical surface, disc, and cylinder, while upper and lower bounds for the order are found for sphere and ball. Improving recent results of Bajnok and Rabau, the authors describe so-called spherical t-designs (Chebyshev-type quadrature formulas of degree t for the sphere with distinct nodes) consisting of O(t3) points.