Norm Inequalities for Derivatives and Differences
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 calcu
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ziirich F. Takens, Groningen
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Man Kam Kwong Anton Zettl
N arm Inequalities for Derivatives and Differences
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Authors Man Kam Kwong Mathematics and Computer Science Division Argonne National Laboratory Argonne, IL 60439 USA Anton Zettl Department of Mathematical Sciences Northern Illinois University DeKalb, IL 60115 USA
Mathematics Subject Classification (1991): 2602, 26D I5, 3902, 39A 12, 39A70, 39B72, 47A30, 47B39
ISBN 3-540-56387-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-56387-3 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1992 Printed in Germany Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 46/3140-543210 - Printed on acid-free paper
Contents
Introd uction
1
1
3
2
Unit Weight Functions 1.1
The Norms of
y
and
1.2
The Norms of
y, y(k),
1.3
Inequalities of Product Form
12
1.4
Growth at Infinity .
26
1.5
Notes and Problems
29
3
y(n)
and
y(n)
6
The Norms of y, y',v"
35
2.1
Introduction.
35
2.2
The Loo Case
35
2.3
The L2 Case
36
2.4
Equivalent Bounded Interval Problems for R
38
2.5
Equivalent Bounded Interval Problems for R+ .
43
2.6
The L1 Case
...................
45
2.7
Upper and Lower Bounds for k(p, R) and k(p, R+)
47
2.8
Extremals . . . . . . . . . . .
55
2.9
Continuity as a Function of p
77
2.10 Landau's Inequality for Nondifferentiable Functions.
80
VI
3
4
2.11 Notes and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
81
Weights
84
3.1 Inequalities of the Sum Form
84
3.2 Inequalities of Product Form
95
3.3 Monotone Weight Functions .
97
3.4 Positive Weight Functions
104
3.5 Weights with Zeros .
109
3.6 Notes and Problems
115
The Difference Operator
117
4.1 The Discrete Product Inequality
117
4.2 The Second Order Case
120
4.3 Extremals . . . . . .
123
4.4 Notes and Problems
137
References
144
Subject Index
149
Preface Edmund Landau's 1913 paper "Einige Ungleichungen fiir zweimal differenzierbare Funktionen", based on earlier work of Hardy and Littlewood, initiated a vast and fruitful research activity involving the study of the relationship between the norms of (i) a function and its derivatives and (ii) a sequence and its differences. These
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