Vol. 67, No. 1, 1976

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Rational approximation to xn

Donald J. Newman and A. R. Reddy

Vol. 67 (1976), No. 1, 247–250
Abstract

This note is concerned with the approximations of xn on [0,1] by polynomials and rational functions having only non-negative coefficients and of degree at most k (1 k n 1). It is shown that the best approximating polynomial of degree k on [0,1] to xn is of the form

pk(x) = dxk,

where d > 0 and satisfies the assumption that

n(1− d) = (n − k)(kn)k∕(n−k)dn∕(n−k),

with an error 𝜖k = 1 d, for each fixed k = 1,2,3,,n 1. It is also shown that dxk is a best approximating rational function of degree k to xn on [0,1].

Mathematical Subject Classification 2000
Primary: 41A20
Milestones
Received: 26 May 1976
Revised: 21 July 1976
Published: 1 November 1976
Authors
Donald J. Newman
A. R. Reddy