Imbedding theorems and inequalities in various metrics for best approximations

V. I. Kolyada

Sbornik. Mathematics, Tome 31 (1977) no. 2, pp. 171-189

Résumé

Let $1\leqslant p\infty$, and let $\lambda=\{\lambda_n\}$ be a sequence of positive numbers with $\lambda_n\downarrow0$. Denote by $E_p(\lambda)$ the class of all functions $f\in L^p(0,2\pi)$ for which the best approximation by trigonometric polynomials satisfies the condition $E_n^{(p)}(f)=O(\lambda_n)$. In this paper the relation between best approximations in different metrics is studied. Necessary and sufficient conditions are found for the imbedding $E_p(\lambda)\subset E_q(\mu)$ ($1$), where $\{\lambda_n\}$ and $\{\mu_n\}$ are positive sequences with $\lambda_n\downarrow0$ and $\mu_n\downarrow0$. Furthermore, it is proved that the condition of P. L. Ul'yanov $$ \sum_{n=1}^\infty n^{q/p-2}\lambda_n^q\infty\qquad(1\leqslant p\infty) $$ is not only sufficient but is also necessary for the imbedding $E_p(\lambda)\subset L^q(0,2\pi)$. The question of imbedding $E_p(\lambda)$ in the space of continuous functions is also considered. Bibliography: 7 titles.