On the \(L_1\)-condition number of the univariate Bernstein basis
Let \[ B^n_j(x):=\left(\begin{matrix} n\\ j\end{matrix} \right) \left({1+x \over 2}\right)^j \left({1-x\over 2} \right)^{n-j}, \quad j=0,\dots,n \] denote the Bernstein basis of degree \(n\) relative to the interval \([-1,1]\). Setting \[ \tau_{n,p}: ={\Bigl\|\sum^n_{j=0} c_jB_j^n\Bigr \|_p\over \|(c_j) \|_p} \] where \((c_j)\) is a \((n+1)\)-tuple and where the norms are the corresponding \(p\)-norms in\break \(L_p[-1,1]\) and \(\mathbb{R}^{n+1}\), respectively, the \(p\)-norm condition number of the Bernstein basis is defined by \(\kappa_{n,p}: = \sup_{(c_j) \neq 0}\tau_{n,p}\sup_{(c_j) \neq 0}\tau^{-1}_{n,p}\). In the main result, the authors show that \[ \lim_{n\to\infty} {\sqrt n\over 2^n} \kappa_{n,1} =\sqrt\pi. \] Using the known result \(\kappa_{n,\infty}=O (2^n)\) and interpolation in the sense of the Riesz-Thorin theorem, this leads to \(\kappa_{n,p}= O(2^n/n^{1/(2p)})\) for \(1\leq p\leq\infty\). According to known lower estimates, this order turns out to be asymptotically correct.
scientific article; zbMATH DE number 1885733