Relationships between the best approximations in different metrics (Q1081774)

The following problem is solved: Let \(L_ p\) \((1\leq p<\infty)\) be a space of \(2\pi\)-periodical summable functions, \(\| f\|_ p=\{\int^{2\pi}_{0}| f(x)|^ p dx\}^{1/p}<\infty\) and let \(E_ n(f)_ p\) be a best approximation by trigonometric polynomials with order not exceeding n. For all \(\epsilon \in H=\{\{\epsilon_ n\}\); \(\epsilon_ n\searrow \emptyset \}\) denote \(L^*_ p(\epsilon)=\{f\in L_ pE_ n(f)_ p=O(\epsilon_ n)\}\). The author gives necessary and sufficient conditions for \(\epsilon\), \(\delta\in H\); \(1\leq p<q<\infty\) \(L^*_ p(\epsilon)\subset L^*_ q(\delta)\) to hold.