Geodesic
Imbedding theorems and inequalities in various metrics for best approximations
Sbornik. Mathematics, Tome 31 (1977) no. 2, pp. 171-189 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $$ \sum_{n=1}^\infty n^{q/p-2}\lambda_n^q<\infty\qquad(1\leqslant p<q<\infty) $$ 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. 
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     author = {V. I. Kolyada},
     title = {Imbedding theorems and inequalities in various metrics for best approximations},
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V. I. Kolyada. Imbedding theorems and inequalities in various metrics for best approximations. Sbornik. Mathematics, Tome 31 (1977) no. 2, pp. 171-189. http://geodesic.mathdoc.fr/item/SM_1977_31_2_a3/

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