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Let p n be a polynomial of m variables and total degree n such that & p n & C(K) =1, where K/R m is a convex body. In this paper we discuss some local and uniform estimates for the magnitude of grad p n under the above conditions. 1999 Academic Press Key Words: multivariate polynomials; convex bodies; gradient and directional derivative of polynomials.
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)
Dedicated with much admiration to Academician Borislav Bojanov on the occasion of his 60th birthday We construct explicitly an extended cubature of Turán type (0, 2) for the unit ball in n. It is a formula for approximation of the integral over the ball by a linear combination of surface integrals over m concentric spheres, centered at the origin, of the… (More)