On rational approximation to \(| x|\) (Q909170)

The author constructs real polynomials \(P_ n(x)\), \(Q_ n(x)\) of degree \(\leq n\), so that \(\| | x| -p(x)/Q(x)\| \leq (2n^ 2)^{- 1}\) where \(\|\) \(\|\) is the uniform norm on [-1,1]. The polynomials p(x) and Q(x) are connected by the relation \(T_{2n}(x^{1/2})=P(x)- xQ(x),\) where \(T_{2n}(x)\) denotes the Chebyshev polynomials (first kind) of degree 2n [see \textit{P. J. Davis}, Interpolation and Approximation (1975; Zbl 0329.41010)].