S. De Marchi

Polynomials arising in factoring generalized Vandermonde determinants II: A condition for monicity

Abstract In our previous paper [1], we observed that generalized Vandermonde determinants of the form V n ; ν ( x 1 ,…, x s ) = | x i μ k |, 1 ≤ i , k ≤ n , where the x i are distinct points belonging to an interval [ a, b ] of the real line, the index n stands for the order, the sequence μ consists of ordered integers 0 ≤ μ 1 2 n , can be factored as a product of the classical Vandermonde determinant and a Schur function . On the other hand, we showed that when x = x n , the resulting polynomial in x is a Schur function which can be factored as a two-factors polynomial: the first is the constant ∏ i =1 n −1 x i μ 1 times the monic polynomial ∏ i =1 n −1 ( x − x i , while the second is a polynomial P M ( x ) of degree M = m n −1 − n + 1. In this paper, we first present a typical application in which these factorizations arise and then we discuss a condition under which the polynomial P M ( x ) is monic .

An interpolant defined by subdivision and the analysis of the error

Given a set of points xi, i=0,...,n on [- 1,1] and the corresponding values yi, i=0,...,n of a 2-periodic function y(x), supplied in some way by interpolation or approximation, we describe a simple method that by doubling iteratively this original set, produces in the limit a smooth function. The analysis of the interpolation error is given.We show that if y ∈ C4 then the error in the p-norm, p = 1, 2 and ∞ depends on the magnitude of the fourth derivative of the function y(x) and on a function α(x) which is even, concave and bounded on [ - 1,1].