Abstract:
Let 1⩽p<∞, and let λ={λn} be a sequence of positive numbers with λn↓0. Denote by Ep(λ) the class of all functions f∈Lp(0,2π) for which the best approximation by trigonometric polynomials satisfies the condition E(p)n(f)=O(λn).
In this paper the relation between best approximations in different metrics is studied. Necessary and sufficient conditions are found for the imbedding Ep(λ)⊂Eq(μ) (1<p<q<∞), where {λn} and {μn} are positive sequences with λn↓0 and μn↓0.
Furthermore, it is proved that the condition of P. L. Ul'yanov
∞∑n=1nq/p−2λqn<∞(1⩽p<q<∞)
is not only sufficient but is also necessary for the imbedding Ep(λ)⊂Lq(0,2π).
The question of imbedding Ep(λ) in the space of continuous functions is also considered.
Bibliography: 7 titles.
\Bibitem{Kol77}
\by V.~I.~Kolyada
\paper Imbedding theorems and inequalities in various metrics for best approximations
\jour Math. USSR-Sb.
\yr 1977
\vol 31
\issue 2
\pages 171--189
\mathnet{http://mi.mathnet.ru/eng/sm2648}
\crossref{https://doi.org/10.1070/SM1977v031n02ABEH002297}
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Linking options:
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https://doi.org/10.1070/SM1977v031n02ABEH002297
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