Moduli of smoothness
Moduli of smoothness
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DOI：
10.1007/978-1-4612-4778-4
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发表时间：
1987
期刊：
影响因子：
--
通讯作者：
Z. Ditzian;V. Totik
中科院分区：
文献类型：
--
作者：
Z. Ditzian;V. Totik
这本书的主题是介绍和应用一种新的函数光滑度度量。虽然我们之前都在这个方向上发表了一些文章，但这里给出的结果是新的。许多工作是1984年夏天在埃德蒙顿完成的，当时我们整合了早期的想法，并制定了文本的大部分细节。又花了一年半的时间来改进和完善许多定理。我们对保罗·内威和理查德·瓦尔加的鼓励表示感谢。我们感谢加拿大NSERC的宝贵支持。我们也感谢克里斯汀·费舍尔和劳拉·海兰仔细地打字我们的手稿。Z.Ditzian V.Totik Contents Introduction.................................................1.第一部分：光滑模第一章Preliminaries......................................7 1.1.Notations................................................7 1.2.关于cp(X).8.1.3.各种步长加权函数cp(X)的例子.第二章.K-泛函和连续模......10 2.1。等价定理.10.2.2.上限估计，Kr.Tp(f，tr)p~mw；(f，t)p，情况I.12...2.3.K-泛函的上界估计，其他情况......16.2.4.K-泛函的下界估计..20.....第三章：K-泛函与光滑模，其他形式。24 3.1.修正的K-泛函..24.3.2.向前和向后的差异......................26......3.3.主件光滑度模数28......
The subject of this book is the introduction and application of a new measure for smoothness offunctions. Though we have both previously published some articles in this direction, the results given here are new. Much of the work was done in the summer of 1984 in Edmonton when we consolidated earlier ideas and worked out most of the details of the text. It took another year and a half to improve and polish many of the theorems. We express our gratitude to Paul Nevai and Richard Varga for their encouragement. We thank NSERC of Canada for its valuable support. We also thank Christine Fischer and Laura Heiland for their careful typing of our manuscript. z. Ditzian V. Totik CONTENTS Introduction................................................. 1 PART I. THE MODULUS OF SMOOTHNESS Chapter 1. Preliminaries...................................... 7 1.1. Notations................................................ 7 1.2. Discussion of Some Conditions on cp (x)....•.......•..•..••. 8...•. 1.3. Examples of Various Step-Weight Functions cp (x)..•..•..•..•... 9..• Chapter 2. The K-Functional and the Modulus of Continuity....... 10 2.1. The Equivalence Theorem........................... 10......... 2.2. The Upper Estimate, Kr. tp (f, tr) p~ Mw;(f, t) p, Case I............ 12... 2.3. The Upper Estimate of the K-Functional, The Other Cases.......... 16. 2.4. The Lower Estimate for the K-Functional................... 20..... Chapter 3. K-Functionals and Moduli of Smoothness, Other Forms. 24 3.1. A Modified K-Functional.......................... 24.......... 3.2. Forward and Backward Differences...................... 26....... 3.3. Main-Part Modulus of Smoothness...................... 28.......