The Direct Theorem of the Theory of Approximation of Periodic Functions with Monotone Fourier Coefficients in Different Metrics


如何引用文章

全文:

开放存取 开放存取
受限制的访问 ##reader.subscriptionAccessGranted##
受限制的访问 订阅存取

详细

n−(l−σ)(∑v=1nvp(l−σ)−1EV−1P)1/p≍(∑v=n+1∞vqσ−1ωlq(f;π/v)p)1/q,n∈N,\({n^{ - (l - \sigma )}}{(\sum\limits_{v = 1}^n {{v^{p(l - \sigma ) - 1}}E_{V - 1}^P} )^{1/p}}\asymp{(\sum\limits_{v = n + 1}^\infty {{v^{q\sigma - 1}}\omega _l^q{{(f;\pi /v)}_p}} )^{1/q}},n \in N,\)
.

In the lower bound in equality (a), the second term nσωl(f; π/n)p generally cannot be omitted. However, if the sequence {ωl(f; π/n)p}n=1 or the sequence {En−1(f)p}n=1 satisfies Bari’s (Bl(p))-condition, which is equivalent to Stechkin’s (Sl)-condition, then

\(E_{n-1}(f)_q\asymp(\sum_{\nu=n+1}^\infty \nu^{q\sigma-1}\omega_l^q (f; \pi/\nu)_{p})^{1/q}, n\in \mathbb{N}.\)

The upper bound in equality (b), which holds for every function \(f \in L_p(\mathbb{T})\) if the series converges, is a strengthened version of the direct theorem. The order equality (b) shows that the strengthened version is order-optimal on the whole class \(M_p(\mathbb{T})\).

作者简介

N. Il’yasov

Baku State University

编辑信件的主要联系方式.
Email: niyazi.ilyasov@gmail.com
阿塞拜疆, Baku, 1148

补充文件

附件文件
动作
1. JATS XML

版权所有 © Pleiades Publishing, Ltd., 2018