Extremal problems for numerical series (Q941898)

The author considers the extremal problem for numerical positive series. That is to find the supremum of the following series \[ \sup_{\forall \{m_k\}} \bigg(\sum_{k=1}^{\infty} \alpha _km_k^r- \sup_{\gamma _n\in\Gamma_n}\;sum_{i\in\gamma_n} \sum_{k\in\Delta_i} \alpha_km_k^r,\;n\in\mathbb N\bigg), \] where \(0<\alpha_k\rightarrow 0\), \(\sum_{k=1}^{\infty}m_k\leq 1\) \((m_k\geq 0)\), \(\{\Delta_i\}\) is a fixed partition of \(\mathbb N\), \(r>0\), \(\Gamma_n\) is the set of all collections of \(\gamma_n\) of \(n\) different natural numbers. As one of the possible applications of the result obtained, he finds solutions of some extremal problems related to best \(n\)-term approximations of periodic functions.