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Analysis and Probability

Wavelets, Signals, Fractals

  • Textbook
  • © 2006

Overview

Authors:
  1. Palle E.T. Jorgensen

    1. Mathematics Department, The University of Iowa, Iowa City, USA

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  • Combines analysis and tools from probability, harmonic analysis, operator theory, and engineering (signal/image processing)
  • Interdisciplinary focus with hands-on approach, generous motivation and new pedagogical techniques
  • Numerous exercises reinforce fundamental concepts and hone computational skills
  • Separate sections explain engineering terms to mathematicians and operator theory to engineers
  • Examines modern trends
  • Fills a gap in the literature

Part of the book series: Graduate Texts in Mathematics (GTM, volume 234)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xlvii
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  2. Introduction: Measures on path space

    Pages 1-37

  3. Transition probabilities: Random walk

    Pages 39-58

  4. ℕo vs. ℤ

    Pages 59-67

  5. A case study: Duality for Cantor sets

    Pages 69-81

  6. Infinite products

    Pages 83-98

  7. The minimal eigenfunction

    Pages 99-108

  8. Generalizations and applications

    Pages 109-156

  9. Pyramids and operators

    Pages 157-177

  10. Pairs of representations of the Cuntz algebras \( \mathcal{O}_n \), and their application to multiresolutions

    Pages 179-204

  11. Back Matter

    Pages 205-276
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About this book

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. —John von Neumann While this is a course in analysis, our approach departs from the beaten path in some ways. Firstly, we emphasize a variety of connections to themes from neighboring fields, such as wavelets, fractals and signals; topics typically not included in a gradu­ ate analysis course. This in turn entails excursions into domains with a probabilistic flavor. Yet the diverse parts of the book follow a common underlying thread, and to­ gether they constitute a good blend; each part in the mix naturally complements the other. In fact, there are now good reasons for taking a wider view of analysis, for ex­ ample the fact that several applied trends have come to interact in new and exciting ways with traditional mathematical analysis—as it was taught in graduate classes for generations. One consequence of these impulses from "outside" is that conventional boundaries between core disciplines in mathematics have become more blurred. Fortunately this branching out does not mean that students will need to start out with any different or additional prerequisites. In fact, the ideas involved in this book are intuitive, natural, many of them visual, and geometric. The required background is quite minimal and it does not go beyond what is typically required in most graduate programs.

Authors and Affiliations

  • Mathematics Department, The University of Iowa, Iowa City, USA

    Palle E.T. Jorgensen

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