Remainder estimates for the approximation numbers of weighted Hardy operators acting onL 2期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Remainder estimates for the approximation numbers of weighted Hardy operators acting onL 2

Authors:	D E Edmunds R Kerman J Lang

Institution:	(1) Present address: Mathematics Institute of the Academy of Sciences of the Czech Republic, Czech;(2) Centre for Mathematical Analysis and its Applications, University of Sussex, Falmer, BN1 9QH Brighton, UK;(3) Department of Mathematics, Brock University, 500 Glendridge Avenue, L2S 3A1 St. Catharines, Ontario, Canada;(4) Mathematics Department, 202 Mathematical Sciences Bldg, 65211 Columbia, MO, USA

Abstract:	We consider the weighted Hardy integral operatorT:L ²(a, b) →L ²(a, b), −∞≤a<b≤∞, defined by $(Tf)(x) = v(x)\smallint _a^x u(t)f(t)dt$ . In EEH1] and EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa _n(T) ofT. In this paper, we show that under suitable conditions onu andv, $\begin{gathered} \mathop {\lim \sup }\limits_{n \to \infty } n^{1/2} \left\| {\frac{1}{\pi }\int {_a^b \left\| {u(t)v(t)} \right\|dt - na_n (T)} } \right\| \hfill \\ \leqslant c(\left\\| {u'} \right\\|_{2/3} + \left\\| {v'} \right\\|_{2/3} )(\left\\| u \right\\|_2 + \left\\| v \right\\|_2 ) + \frac{3}{\pi }\left\\| {uv} \right\\|_1 , \hfill \\ \end{gathered}$ where ∥w∥_p=(∫ _a ^b \|w(t)\|^p dt)^1/p. Research supported by NSERC, grant A4021. Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic.

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