Markov- and Bernstein-type inequalities for Müntz polynomials and exponential sums in \(L_p\)
The main result of this paper is the following Markov-type inequality for Müntz polynomials. Let \(\Lambda=(\lambda_j)_{j=0}^\infty\) be an increasing sequence of nonnegative real numbers. Suppose \(\lambda_0=0\) and there exists a \(\delta>0\) so that \(\lambda_j\geq\delta j\) for each \(j\). Suppose \(0<a<b\) and \(1\leq p\leq\infty\). Then there exists a constant \(c(a,b,\delta)\) depending only on \(a\), \(b\), and \(\delta\) so that \(\|P^\prime\|_{L_p[a,b]}\leq c(a,b,\delta)(\sum_{j=0}^n\lambda_j)\|P\|_{L_p[a,b]}\) for every \(P\in M_n(\Lambda)\), the linear span of \(\{x^{\lambda_0},x^{\lambda_1},\dots,x^{\lambda_n}\}\) over \(\mathbb{R}\). This is an \(L_p\) analogue of an earlier result of \textit{P. B. Borwein} and \textit{T. Erdélyi} [J. Approximation Theory 85, No. 2, 132-139(1996; Zbl 0853.41008)] establishing such an inequality with respect to the sup norm on \([a,b]\) with \(b>a>0\). The origin of Markov-type inequalities for Müntz polynomials is a well-known Newman inequality [\textit{D. J. Newman}, J. Approximation Theory 18, 360-362 (1976; Zbl 0441.41002)] proved in the case where \([a,b]=[0,1]\), which plays a special role in this theory. Let us note that analogues of this result on \([a,b]\), \(a>0\), cannot be obtained by a simple transformation \(y=\alpha x+\beta\) which does not preserve membership in \(M_n(\Lambda)\) in general.
scientific article; zbMATH DE number 1455762
scientific article