On a generalization of the Bessel inequality
Let \(\{\varphi_ k(x)\}^ n_{i= 1}\) and \(\{\psi_ i(x)\}^ s_{i= 1}\) be orthonormal systems on the interval \([0,1]\) \((s\leq n)\) and \(\{a_ i\}^ s_{i= 1}\) a sequence of numbers. We put \[ f_ r(x)= \sum^ r_{i= 1} a_ i \psi_ i(x),\quad r= 1,2,\dots, s,\quad\text{and } (f,\varphi)= \int^ 1_ 0 f(x) \varphi(x) dx. \] Let \(r= r(k)\) be a monotone increasing function taking natural values \((r\leq s)\), and we denote by \(\widetilde s\) the smallest number of the form \(2^ l\) which is larger than \(s\). The following inequality holds: \[ \sum^ n_{k= 1} (f_ r, \varphi_ k)^ 2\leq (\log_ 2 \widetilde s+ 1)^ 2 \sum^ s_{i= 1} a^ 2_ i. \] {}.
scientific article; zbMATH DE number 525683