QUADRATURE-RULES FOR THE SURFACE INTEGRAL OF THE UNIT-SPHERE BASED ON EXTREMAL FUNDAMENTAL SYSTEMS

Quadrature rules for the surface integral of the unit sphere S-r-1 based on an extremal fundamental system, i.e., a nodal system which provides fundamental Lagrange interpolatory polynomials with minimal uniform norm, are investigated. Such nodal systems always exist; their construction has been given in earlier work. Here the main results is that the corresponding interpolatory quadrature for the space of homogeneous polynomials of degree two is equally weighted for arbitrary r, and hence positive. For the full quadratic polynomial space we can prove positivity of the weights, only.