On certain extremal problems (Q1088104)

The following problem is considered: given a linearly independent system \(\{x_ j\}^ n_ 1\) in a Hilbert space, find the largest value of \(\max_{1\leq k\leq n}| d_ k|\) under the condition that \(\| d_ 1x_ 1+...+d_ nx_ n\| \leq 1\). For certain classical systems of rational fractions, exponentials, etc. the authors find this largest value and give explicit formulae for the extremal 'polynomial' \(d_ 1x_ 1+...+d_ nx_ n\). (For example, the system \(\{(\exp (-\lambda_ jt))t^{s_ j-1}\}\quad 1\leq j\leq n\) in \(L^ 2({\mathbb{R}}_+)\) is considered, where Re\(\lambda_ j>0\) and \(s_ k=card \{i\leq k:\lambda_ i=\lambda_ k\}.)\)