On multiple extremal problems

Abstract:

In linear space $X$ consider arbitrary norms $\|\cdot \|_j, j=1,2,...,d,$ and linear subspaces $M_j\subset X$ of dim $M_j=n_j, j=1,2,...,d,$ and set $n=n_1+\dots +n_d$. Then given any subspace $U\subset X$ of dimension $n+1$ we shall verify the existence of $u\in U\setminus \{0\}$ such that for every $1\leq j\leq d$ the element $u$ is orthogonal to $M_j$ in the $\|\cdot \|_j$ norm, that is, \[ \|u\|_j\leq \|u-m\|_j,\;\; \forall m\in M_j, \forall 1\leq j\leq d.\] In case of polynomial approximation with respect to distinct weighted uniform norms, the above extremal element is called a multiple Chebyshev polynomial. It will be shown that for weights whose ratios are monotone there always exists a unique multiple Chebyshev polynomial of maximal degree.