On the approximation of holomorphic functions by Müntz polynomials on an interval away from the origin (Q1067135)

Let \((\lambda_{\nu})^{\infty}_{\nu =0}\) be a given sequence of numbers, \(0\leq \lambda_ 0<\lambda_ 1<...\to \infty\) and let \(\Pi_ n(\lambda_{\nu})\) (n\(\in {\mathbb{N}}\) fixed) denote the set of polynomials \(p_ n(x)=\sum^{\infty}_{\nu =0}a_{\nu}x^{\lambda_{\nu}}.\) The problem of the approximation of functions by polynomials \(p_ n\) on an interval [a,b] with \(a>0\) in the Chebyshev-norm is considered. It is shown that, in contrast to the approximation on an interval [0,b] for the approximation of a function f by polynomials \(p_ n\) on [a,b], \(a>0\), the minimal deviation \(d_ n(f):=\min_{p_ n\in \Pi_ n(\lambda_{\nu})}\| f-p_ n\|_{[a,b]}\) tends to zero with a geometric rate (as \(n\to \infty)\) for all functions f holomorphic in a sufficiently large region around the interval [a,b]. Also converse theorems are given. The results are interpreted in the context of the equivalent (linear) problem of the approximation of functions by exponential sums.