Analysis in Integer and Fractional Dimensions available in Hardcover
- Pub. Date:
- Cambridge University Press
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. Topics include the (two-dimensional) Grothendieck inequality and its extensions to higher dimensions, stochastic models of Brownian motion, degrees of randomness and Fréchet measures in stochastic analysis. This book is primarily aimed at graduate students specializing in harmonic analysis, functional analysis or probability theory. It contains many exercises and is suitable as a textbook. It is also of interest to computer scientists, physicists, statisticians, biologists and economists.
Table of Contents
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.