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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)
This is the story of the classical Markov inequality for the k-th derivative of an algebraic polynomial, and of the remarkably many attempts to provide it with alternative proofs that occurred all through the last century. In our survey we inspect each of the existing proofs and describe, sometimes briefly, sometimes not very briefly, the methods and ideas… (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)