Norm Inequalities for Derivatives and Differences / Edition 1

Norm Inequalities for Derivatives and Differences / Edition 1

by Man K. Kwong, Anton Zettl
     
 

Norm inequalities relating (i) a function and two of its derivatives and (ii) a sequence and two of its differences are studied. Detailed elementary proofs of basic inequalities are given. These are accessible to anyone with a background of advanced calculus and a rudimentary knowledge of the Lp and lp spaces.
The classical inequalities associated with the names

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Overview

Norm inequalities relating (i) a function and two of its derivatives and (ii) a sequence and two of its differences are studied. Detailed elementary proofs of basic inequalities are given. These are accessible to anyone with a background of advanced calculus and a rudimentary knowledge of the Lp and lp spaces.
The classical inequalities associated with the names of
Landau, Hadamard, Hardy and Littlewood, Kolmogorov,
Schoenberg and Caravetta, etc., are discussed, as well as their discrete analogues and weighted versions. Best constants and the existence and nature of extremals are studied and many open questions raised. An extensive list of references is provided, including some of the vast Soviet literature on this subject.

Product Details

ISBN-13:
9783540563877
Publisher:
Springer Berlin Heidelberg
Publication date:
03/05/1993
Series:
Lecture Notes in Mathematics Series, #1536
Edition description:
1992
Pages:
152
Product dimensions:
9.21(w) x 6.14(h) x 0.34(d)

Table of Contents

Introduction1
1Unit Weight Functions3
1.1The Norms of y and y[superscript (n)]3
1.2The Norms of y, y[superscript (k)], and y[superscript (n)]6
1.3Inequalities of Product Form12
1.4Growth at Infinity26
2The Norms of y, y', y''35
2.2The [actual symbol not reproducible] Case35
2.3The L[superscript 2] Case36
2.4Equivalent Bounded Interval Problems for R38
2.5Equivalent Bounded Interval Problems for R[superscript +]43
2.6The L[superscript 1] Case45
2.7Upper and Lower Bounds for k(p, R) and k(p, R[superscript +])47
2.8Extremals55
2.9Continuity as a Function of p77
2.10Landau's Inequality for Nondifferentiable Functions80
3Weights84
3.1Inequalities of the Sum Form84
3.2Inequalities of Product Form95
3.3Monotone Weight Functions97
3.4Positive Weight Functions104
3.5Weights with Zeros109
4The Difference Operator117
4.1The Discrete Product Inequality117
4.2The Second Order Case120
4.3Extremals123
References144
Subject Index149

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