A Bernstein-Markov theorem for normed spaces
For positive integers \(k\) and \(m\), with \(0\leq k\leq m\), let \(c_{m,k}\equiv \sup\{| p^{(k)}(0)|\): \(p\) is a polynomial satisfying \(| p(t)|\leq (1+| t|)^m\), \(t\in \mathbb{R}\}\). The author's main result is the following: Theorem: Let \(X\) and \(Y\) be normed spaces over \(\mathbb{K}= \mathbb{R}\) or \(\mathbb{C}\), and let \(\phi:X\to \mathbb{R}_+\) satisfy \(\phi(x+ y)\leq \phi(x)+\| y\|\) for every \(x,y\in X\). Let \(P: X\to Y\) be a polynomial satisfying \(\| P(x)\|\leq \phi(x)^m\) for all \(x\in X\). Then, if \(\mathbb{K}= \mathbb{R}\), \(\|\widehat D^kP(x)\|\leq c_{m,k}\phi(x)^{m- k}\), while if \(\mathbb{K}= \mathbb{C}\), \[ \|\widehat D^kP(x)\|\leq {m^mk!\over k^k(m- k)^{m- k}} \phi(x)^{m- k}. \] As the author notes, the constant \(c_{m,k}\) is best possible even when \(X= \mathbb{R}\). With \(\phi(x)= \| x\|\), the author obtains as a consequence the following result: Theorem: For any real normed spaces \(X\) and \(Y\) and any polynomial \(P:X\to Y\) of degree \(m\), \(\|\widehat D^kP\|\leq c_{m,k}\| P\|\) \((0\leq k\leq m)\). It follows that if \(X\) and \(Y\) are real and \(F:X\times\cdots \times X\to Y\) is a continuous symmetric \(m\)-linear mapping with associated polynomial \(\widehat F\), then for any vectors \(x_1,\dots, x_n\in X\) and any nonnegative integers \(k_1,\dots, k_n\) such that \(k_1+\cdots+ k_n= m\), \(\| F(x^{k_1}_1,\dots, x^{k_n}_n)\|\leq \sqrt{{m^m\over k^{k_1}_1\cdots k^{k_n}_n}} \| \widehat F\|\). Note that the problem of finding the best constant in the previous inequality is open. Finally, the author investigates the problem of estimating the value of \(c_{m,k}\), work which was continued in the author's article in J. Approx. Theory 93, 293-312 (1998).
scientific article; zbMATH DE number 1031584