The purpose of this paper is to show certain links between uni-variate interpolation by algebraic polynomials and the presentation of poly-harmonic functions. This allows us to construct cubature formulae for multi-variate functions having highest order of precision with respect to the class of polyharmonic functions. We obtain a Gauss type cubature formula… (More)
A polynomial of degree n in two variables is shown to be uniquely determined by its Radon projections taken over [n/2] + 1 parallel lines in each of the (2[(n + 1)/2] + 1) equidistant directions along the unit circle.
We discuss quadrature formulae of highest algebraic degree of precision for integration of functions of one or many variables which are based on non-standard data, i.e., in which the information used is different from the standard sampling of function values. Among the examples given in this survey is a quadrature formula for integration over the disk,… (More)
We give a bivariate analog of the Micchelli-Rivlin quadrature for computing the integral of a function over the unit disk using its Radon projections.
We construct a formula for numerical integration of functions over the unit ball in R d that uses n Radon projections of these functions and is exact for all algebraic polynomials in R d of degree 2n&1. This is the highest algebraic degree of precision that could be achieved by an n term integration rule of this kind. We prove the uniqueness of this… (More)