On a result by Geronimus
Let \(E_n(f)_L\) be best approximation in \(L(\pi)\) of \(2\pi\)-periodic function \(f\) from this space. The authors propose new proof the next \textit{J. Geronimus} theorem [Ann. Math. (2) 37, 483--517 (1936; Zbl 0015.01401)]. Let \(f_{n,q}(t):= \sin(n+1)t- 2q\sin t\), \(t\in\pi\), where \(n\in\mathbb{N}\), \(t\in\mathbb{R}\). Then for \(|q|\geq 1\), \(E_{n-1}(f_{n,q})_L= 8|q|\) and for \(|q|< 1\) equals \(4(1+ q)^2\).
scientific article; zbMATH DE number 5966627