On the best conditioned bases of quadratic polynomials
Let \(u=(u_1,\dots,u_n)\) be a basis of a subspace \(U\) in a Banach space \(X\). The \(\ell_\infty\)-condition number of \(u\) is the number \[ K_\infty(u)= \sup_{a\in \ell^n_\infty}\frac {\|\sum^n_1a_iu_i \|} {\| a\|_\infty} \sup_{a\in\ell^n_\infty} \frac{\| a \|_\infty} {\|\sum^n_1a_iu_i \|}. \] The author finds a basis \(u^0\) of the space of polynomials of degree two \({\mathcal P}_2\Subset[-1,1]\) such that \(K_\infty(u^0)\approx 1.24839\). For arbitrary \(u\) one has in this case \(K_\infty({\mathcal P}_2)\leq 1.25\) and the upper is achieved for some Lagrange basis. The result in particular, means that there is no Lagrange basis for \({\mathcal P}_2\) of the optimal condition number.
scientific article; zbMATH DE number 2107875