Best approximation and interpolation of (1+(ax)2)−1 and its transforms Original Research Article

We show that Lagrange interpolants at the Chebyshev zeros yield best relative polynomial approximations of (1+(ax)2)−1 on [−1,1], and more generally of∫0∞ dμ(a)1+(ax)2,where μ is a suitably restricted measure. We use this to study relative approximation of (1+x2)−1 on an increasing sequence of intervals, and Lagrange interpolation of |x|γ. Moreover, we show how it gives a simple proof of identities for some trigonometric sums.