Abstract

We establish sharp order estimates for the error of optimal cubature formulas on the Nikol’skii–Besov and Lizorkin–Triebel type spaces, \(B^{s\,\mathtt{m}}_{p\,q}(\mathbb T^m)\) and \(L^{s\,\mathtt{m}}_{p\,q}(\mathbb T^m)\), respectively, for a number of relations between the parameters \(s\), \(p\), \(q\), and \(\mathtt{m}\) (\(s=(s_1,\dots,s_n)\in\mathbb R^n_+\), \(1\leq p,q\leq\infty\), \(\mathtt{m}=(m_1,\dots,m_n)\in{\mathbb N}^n\), \(m=m_1+\dots+m_n\)). Lower estimates are proved via Bakhvalov’s method. Upper estimates are based on Frolov’s cubature formulas.

中文翻译：

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几类周期函数类别上的最优Cubase公式

摘要

我们针对Nikol'skii–Besov和Lizorkin–Triebel类型空间上的最佳孵化器公式的误差建立了清晰的阶估计，\（B ^ {s \，\ mathtt {m}} _ {p \，q}（\ mathbb T ^ m）\）和\（L ^ {s \，\ mathtt {m}} _ {p \，q}（\ mathbb T ^ m）\）分别表示参数\（ s \），\（p \），\（q \）和\（\ mathtt {m} \）（\（s =（s_1，\ dots，s_n）\ in \ mathbb R ^ n _ + \），\（1 \ leq p，q \ leq \ infty \），\（\ mathtt {m} =（m_1，\ dots，m_n）\ in {\ mathbb N} ^ n \），\（m = m_1 + \ dots + m_n \））。较低的估计值通过Bakhvalov的方法证明。较高的估计值基于Frolov的孵化公式。