-widths of certain function classes defined by the modulus of continuity

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• 作者：Mirgand Shabozovich Shabozov^a ; ^{shabozov@mail.ru} ; Gulzorkhon Amirshoevich Yusupov^b ; ^{G_7777@mail.ru} ; Sofiya Davronbekovna Temurbekova^a ; ^{sofish-83@mail.ru}
• 关键词：Best approximation ; Modulus of continuity ; Differentiable periodic functions ; Inequalities of Jackson&ndash ; Stechkin ; Widths
• 刊名：Journal of Approximation Theory
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：215
• 期：Complete
• 页码：145-162
• 全文大小：305 K
• 卷排序：215

文摘

Several exact inequalities are found between the best approximation by trigonometric polynomials in L2L2 of differentiable periodic functions, in the sense of Weyl, averaged with the weight of the modulus of continuity of arbitrary fractional order. For some classes of functions, defined by specific moduli of continuity, the exact values of the various nn-widths in L2L2 are calculated. In particular the problem of minimizing the constants in inequalities of Jackson&ndash;Stechkin type over all subspaces of dimension NN is solved. It is proved that this value is equal to the exact value of the different widths of the class L2(α)(β,p,h,φ), where φφ is a non-negative integrable weight function on (0,h](0<h≤π/n).