Analysis in Integer and Fractional Dimensions

Thorough and self-contained study for graduate students and researchers.

Ron Blei (Author)

9780521650847, Cambridge University Press

Hardback, published 19 July 2001

580 pages, 275 exercises
23.6 x 16 x 3.3 cm, 0.885 kg

This book provides a thorough and self-contained study of interdependence and complexity in settings of functional analysis, harmonic analysis and stochastic analysis. It focuses on 'dimension' as a basic counter of degrees of freedom, leading to precise relations between combinatorial measurements and various indices originating from the classical inequalities of Khintchin, Littlewood and Grothendieck. The basic concepts of fractional Cartesian products and combinatorial dimension are introduced and linked to scales calibrated by harmonic-analytic and stochastic measurements. Topics include the (two-dimensional) Grothendieck inequality and its extensions to higher dimensions, stochastic models of Brownian motion, degrees of randomness and Frechet measures in stochastic analysis. This book is primarily aimed at graduate students specialising in harmonic analysis, functional analysis or probability theory. It contains many exercises and is suitable to be used as a textbook. It is also of interest to scientists from other disciplines, including computer scientists, physicists, statisticians, biologists and economists.

Preface
1. A prologue: mostly historical
2. Three classical inequalities
3. A fourth inequality
4. Elementary properties of the Frechet variation - an introduction to tensor products
5. The Grothendieck factorization theorem
6. An introduction to multidimensional measure theory
7. An introduction to harmonic analysis
8. Multilinear extensions of the Grothendieck inequality
9. Product Frechet measures
10. Brownian motion and the Wiener process
11. Integrator
12. A '3/2n- dimensional' Cartesian product
13. Fractional Cartesian products and combinatorial dimension
14. The last chapter: leads and loose ends.

Subject Areas: Calculus & mathematical analysis [PBK]