A Remez-type inequality for non-dense Müntz spaces with explicit bound (Q1266103)

A Remez-type inequality for Müntz polynomials \(p_n(x)=\sum_{k=0}^n a_k x^{\lambda_k}\), or equivalently for Dirichlet sums \(P_n(t)=\sum_{k=0}^n a_k e^{-\lambda_kt}\), is established, where \(\Lambda:=(\lambda_k)_{k=0}^\infty\) denotes a sequence of real numbers satisfying \(0=\lambda_0<\lambda_1<\lambda_2<\ldots\) and \(\sum_{k=1}^\infty 1/\lambda_k<\infty\). An explicit upper bound for a certain constant [discussed in the authors' paper, J. Am. Math. Soc. 10, No. 2, 327-349 (1997; Zbl 0864.41014)] that occurs in the Remez-type inequality is given. In the case \(\lambda_k=k^\alpha\), \(\alpha>1\) it is verified that the bounds are essentially sharp.