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Differentiability of Six Operators on Nonsmooth Functions and P-Variation : Lecture Notes in Mathematics - R. M. Dudley

Differentiability of Six Operators on Nonsmooth Functions and P-Variation

Lecture Notes in Mathematics

By: R. M. Dudley, R. Norvaisa, J. Qian (Contribution by)

Paperback Published: July 1999
ISBN: 9783540659754
Number Of Pages: 282

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The book is about differentiability of six operators on functions or pairs of functions: composition (f of g), integration (of f dg), multiplication & convolution of two functions, both varying, & the product integral & inverse operators for one function. The operators are differentiable with respect to p-variation norms with optimal remainder bounds. Thus the functions as arguments of the operators can be nonsmooth, possibly discontinuous, but four of the six operators turn out to be analytic (holomorphic) for some p-variation norms. The reader will need to know basic real analysis, including Riemann & Lebesgue integration. The book is intended for analysts, statisticians & probabilists. Analysts & statisticians have each studied the differentiability of some of the operators from different viewpoints, & this volume seeks to unify & expand their results.

Prefacep. v
A survey on differentiability of six operators in relation to probability and statistics
Contents of Part Ip. 1
Introduction; kinds of differentiabilityp. 2
The two-function composition operator (F,G) &prompto; F&rbull;Gp. 12
The quantile (inverse) operatorp. 17
The integration operator and Young integralsp. 20
The product integralp. 27
Probability and p-vaxiationp. 30
General and C-differentiabilityp. 35
The chain rule and concluding remarksp. 37
Appendicesp. 43
Referencesp. 67
Product integrals, Young integrals and p-variation
Abstractp. 73
Contents of Part IIp. 73
Introductionp. 75
p-variationp. 81
Stieltjes and Young integralsp. 98
Product integralsp. 139
Indefinite product integralsp. 170
The logarithm operatorp. 198
Referencesp. 205
Differentiability of the composition and inverse operators for regulated and a.e. continuous functions
Abstractp. 209
Contents of Part IIIp. 209
Introductionp. 210
Regulated functionsp. 213
Almost everywhere continuous functionsp. 220
The quantile operatorp. 222
The composition operator for real-valued functionsp. 228
The composition operator for Banach-valued functionsp. 234
Appendixp. 236
Referencesp. 237
Bibliographies on p-variation and ¿-variation
Abstractp. 241
Contents of Part IVp. 241
Introductionp. 242
Bibliography on p-variationp. 243
Bibliography on ¿-variationp. 262
Subject Indexp. 273
Author Indexp. 275
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540659754
ISBN-10: 3540659757
Series: Lecture Notes in Mathematics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 282
Published: July 1999
Country of Publication: DE
Dimensions (cm): 23.2 x 15.4  x 1.8
Weight (kg): 0.42