Fourier Analysis and Hausdorff Dimension PDF Download

Author: Pertti Mattila
Publisher: Cambridge University Press
ISBN: 1316352528
Category : Mathematics
Languages : en
Pages : 455

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Book Description
During the past two decades there has been active interplay between geometric measure theory and Fourier analysis. This book describes part of that development, concentrating on the relationship between the Fourier transform and Hausdorff dimension. The main topics concern applications of the Fourier transform to geometric problems involving Hausdorff dimension, such as Marstrand type projection theorems and Falconer's distance set problem, and the role of Hausdorff dimension in modern Fourier analysis, especially in Kakeya methods and Fourier restriction phenomena. The discussion includes both classical results and recent developments in the area. The author emphasises partial results of important open problems, for example, Falconer's distance set conjecture, the Kakeya conjecture and the Fourier restriction conjecture. Essentially self-contained, this book is suitable for graduate students and researchers in mathematics.

Author: Pertti Mattila
Publisher: Cambridge University Press
ISBN: 1107107350
Category : Mathematics
Languages : en
Pages : 455

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Book Description
Modern text examining the interplay between measure theory and Fourier analysis.

Author: Christopher J. Bishop
Publisher: Cambridge University Press
ISBN: 1107134110
Category : Mathematics
Languages : en
Pages : 415

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Book Description
A mathematically rigorous introduction to fractals, emphasizing examples and fundamental ideas while minimizing technicalities.

Author: K. J. Falconer
Publisher: Cambridge University Press
ISBN: 9780521337052
Category : Mathematics
Languages : en
Pages : 184

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Book Description
A mathematical study of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Considers questions of local density, the existence of tangents of such sets as well as the dimensional properties of their projections in various directions.

Author: Thomas H. Wolff
Publisher: American Mathematical Soc.
ISBN: 0821834495
Category : Mathematics
Languages : en
Pages : 154

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Book Description
This book demonstrates how harmonic analysis can provide penetrating insights into deep aspects of modern analysis. It is both an introduction to the subject as a whole and an overview of those branches of harmonic analysis that are relevant to the Kakeya conjecture. The usual background material is covered in the first few chapters: the Fourier transform, convolution, the inversion theorem, the uncertainty principle and the method of stationary phase. However, the choice of topics is highly selective, with emphasis on those frequently used in research inspired by the problems discussed in the later chapters. These include questions related to the restriction conjecture and the Kakeya conjecture, distance sets, and Fourier transforms of singular measures. These problems are diverse, but often interconnected; they all combine sophisticated Fourier analysis with intriguing links to other areas of mathematics and they continue to stimulate first-rate work. The book focuses on laying out a solid foundation for further reading and research. Technicalities are kept to a minimum, and simpler but more basic methods are often favored over the most recent methods. The clear style of the exposition and the quick progression from fundamentals to advanced topics ensures that both graduate students and research mathematicians will benefit from the book.

Author: Pertti Mattila
Publisher: Cambridge University Press
ISBN: 9780521655958
Category : Mathematics
Languages : en
Pages : 360

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Book Description
This book studies the geometric properties of general sets and measures in euclidean space.

Author: Claude Ambrose Rogers
Publisher: Cambridge University Press
ISBN: 9780521624916
Category : Mathematics
Languages : en
Pages : 230

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Book Description
When it was first published this was the first general account of Hausdorff measures, a subject that has important applications in many fields of mathematics. There are three chapters: the first contains an introduction to measure theory, paying particular attention to the study of non-s-finite measures. The second develops the most general aspects of the theory of Hausdorff measures, and the third gives a general survey of applications of Hausdorff measures followed by detailed accounts of two special applications. This edition has a foreword by Kenneth Falconer outlining the developments in measure theory since this book first appeared. Based on lectures given by the author at University College London, this book is ideal for graduate mathematicians with no previous knowledge of the subject, but experts in the field will also want a copy for their shelves.

Author: Loukas Grafakos
Publisher: Springer Science & Business Media
ISBN: 0387094326
Category : Mathematics
Languages : en
Pages : 494

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Book Description
The primary goal of this text is to present the theoretical foundation of the field of Fourier analysis. This book is mainly addressed to graduate students in mathematics and is designed to serve for a three-course sequence on the subject. The only prerequisite for understanding the text is satisfactory completion of a course in measure theory, Lebesgue integration, and complex variables. This book is intended to present the selected topics in some depth and stimulate further study. Although the emphasis falls on real variable methods in Euclidean spaces, a chapter is devoted to the fundamentals of analysis on the torus. This material is included for historical reasons, as the genesis of Fourier analysis can be found in trigonometric expansions of periodic functions in several variables. While the 1st edition was published as a single volume, the new edition will contain 120 pp of new material, with an additional chapter on time-frequency analysis and other modern topics. As a result, the book is now being published in 2 separate volumes, the first volume containing the classical topics (Lp Spaces, Littlewood-Paley Theory, Smoothness, etc...), the second volume containing the modern topics (weighted inequalities, wavelets, atomic decomposition, etc...). From a review of the first edition: “Grafakos’s book is very user-friendly with numerous examples illustrating the definitions and ideas. It is more suitable for readers who want to get a feel for current research. The treatment is thoroughly modern with free use of operators and functional analysis. Morever, unlike many authors, Grafakos has clearly spent a great deal of time preparing the exercises.” - Ken Ross, MAA Online

Author: Antoni Zygmund
Publisher: Princeton University Press
ISBN: 1400881951
Category : Mathematics
Languages : en
Pages : 196

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Book Description
A classic treatment of Fourier analysis from the acclaimed Annals of Mathematics Studies series Princeton University Press is proud to have published the Annals of Mathematics Studies since 1940. One of the oldest and most respected series in science publishing, it has included many of the most important and influential mathematical works of the twentieth century. The series continues this tradition as Princeton University Press publishes the major works of the twenty-first century. To mark the continued success of the series, all books are available in paperback and as ebooks.

Author: Ciprian Demeter
Publisher: Cambridge University Press
ISBN: 1108499708
Category : Mathematics
Languages : en
Pages : 349

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Book Description
Comprehensive coverage of recent, exciting developments in Fourier restriction theory, including applications to number theory and PDEs.