Sidon and Kronecker-like Sets in Compact Abelian Groups

Sidon and Kronecker-like Sets in Compact Abelian Groups PDF Author: Xu Yang
Publisher:
ISBN:
Category : Compact Abelian groups
Languages : en
Pages : 129

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Book Description
Let $G$ be a compact abelian group and $\Ga$ be its discrete dual group. In this thesis we study various types of interpolation sets. A subset $E \subset \Ga$ is Sidon if every bounded function on $E$ can be interpolated by the Fourier transform of a finite complex measure. Sidon sets have been extensively studied, and one significant breakthrough, that Sidonicity is equivalent to proportional quasi-independence, was proved by Bourgain and Pisier during the early 80s. In this thesis we will give a detailed demonstration of Pisier's approach. We also seek for possible extensions of Pisier's theorems. Based on Pisier's techniques, we will show Sidonicity is equivalent to proportional independence of higher degrees and minimal constants. A subset $E \subset \Ga$ is $\e$-Kronecker if every function on $E$ with range in the unit circle can be interpolated by a continuous character on $\Ga$ with an error of $\e$. We will prove some interesting properties of $\e$-Kronecker sets and give an estimation of the Sidon constant of such sets. Generalizations of Kronecker sets include binary Kronecker sets and $N$-pseudo-Rademacher sets. We compute the binary Kronecker constants of some interesting examples. For $N$-pseudo-Rademacher sets, we give a characterization of such sets, describe their structures and prove the existence of large $N$-pseudo-Rademacher sets.

Sidon and Kronecker-like Sets in Compact Abelian Groups

Sidon and Kronecker-like Sets in Compact Abelian Groups PDF Author: Xu Yang
Publisher:
ISBN:
Category : Compact Abelian groups
Languages : en
Pages : 129

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Book Description
Let $G$ be a compact abelian group and $\Ga$ be its discrete dual group. In this thesis we study various types of interpolation sets. A subset $E \subset \Ga$ is Sidon if every bounded function on $E$ can be interpolated by the Fourier transform of a finite complex measure. Sidon sets have been extensively studied, and one significant breakthrough, that Sidonicity is equivalent to proportional quasi-independence, was proved by Bourgain and Pisier during the early 80s. In this thesis we will give a detailed demonstration of Pisier's approach. We also seek for possible extensions of Pisier's theorems. Based on Pisier's techniques, we will show Sidonicity is equivalent to proportional independence of higher degrees and minimal constants. A subset $E \subset \Ga$ is $\e$-Kronecker if every function on $E$ with range in the unit circle can be interpolated by a continuous character on $\Ga$ with an error of $\e$. We will prove some interesting properties of $\e$-Kronecker sets and give an estimation of the Sidon constant of such sets. Generalizations of Kronecker sets include binary Kronecker sets and $N$-pseudo-Rademacher sets. We compute the binary Kronecker constants of some interesting examples. For $N$-pseudo-Rademacher sets, we give a characterization of such sets, describe their structures and prove the existence of large $N$-pseudo-Rademacher sets.

Interpolation and Sidon Sets for Compact Groups

Interpolation and Sidon Sets for Compact Groups PDF Author: Colin Graham
Publisher: Springer Science & Business Media
ISBN: 1461453925
Category : Mathematics
Languages : en
Pages : 262

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Book Description
Understanding special sets of integers was classically of interest to Hadamard, Zygmund and others, and continues to be of interest today. This book is a modern treatment of the subject of interpolation and Sidon sets. It is a unique book, aimed at both new and experienced researchers. In particular, this is the only book in English which features a complete treatment of the Pisier-Bourgain results on Sidon sets, many of which were originally in French, in hard to access publications. Applications of the P-B results, due to Pisier, Bourgain, Ramsey, and the authors are included. The book introduces the reader to a wealth of methods important in mathematics today: topological, probabilistic, algebraic, combinatoric and analytic. It prepares students to perform research in the area and provides both exercises and open problems. The book also provides direction to the literature for topics it does not fully cover. The book is self-contained, with appendices covering results that are required, but not necessarily in the pre-requisite background of a student ready to choose an area for research in harmonic analysis. ​

Sidon Sets Associated with a Closed Subset of a Compact Abelian Group

Sidon Sets Associated with a Closed Subset of a Compact Abelian Group PDF Author: E. V. Dudley
Publisher:
ISBN:
Category : Compact Abelian groups
Languages : en
Pages : 14

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Book Description


Encyclopaedia of Mathematics

Encyclopaedia of Mathematics PDF Author: Michiel Hazewinkel
Publisher: Springer Science & Business Media
ISBN: 9781556080036
Category : Mathematics
Languages : en
Pages : 540

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Book Description
V.1. A-B v.2. C v.3. D-Feynman Measure. v.4. Fibonaccimethod H v.5. Lituus v.6. Lobachevskii Criterion (for Convergence)-Optical Sigman-Algebra. v.7. Orbi t-Rayleigh Equation. v.8. Reaction-Diffusion Equation-Stirling Interpolation Fo rmula. v.9. Stochastic Approximation-Zygmund Class of Functions. v.10. Subject Index-Author Index.

Encyclopaedia of Mathematics

Encyclopaedia of Mathematics PDF Author: M. Hazewinkel
Publisher: Springer
ISBN: 1489937951
Category : Mathematics
Languages : en
Pages : 967

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Book Description


Fourier Analysis on Groups

Fourier Analysis on Groups PDF Author: Walter Rudin
Publisher: Courier Dover Publications
ISBN: 0486821013
Category : Mathematics
Languages : en
Pages : 305

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Book Description
Self-contained treatment by a master mathematical expositor ranges from introductory chapters on basic theorems of Fourier analysis and structure of locally compact Abelian groups to extensive appendixes on topology, topological groups, more. 1962 edition.

Pontryagin Duality and the Structure of Locally Compact Abelian Groups

Pontryagin Duality and the Structure of Locally Compact Abelian Groups PDF Author: Sidney A. Morris
Publisher: Cambridge University Press
ISBN: 0521215439
Category : Mathematics
Languages : en
Pages : 141

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Book Description
These lecture notes begin with an introduction to topological groups and proceed to a proof of the important Pontryagin-van Kampen duality theorem and a detailed exposition of the structure of locally compact abelian groups. Measure theory and Banach algebra are entirely avoided and only a small amount of group theory and topology is required, dealing with the subject in an elementary fashion. With about a hundred exercises for the student, it is a suitable text for first-year graduate courses.

Open Problems in Topology II

Open Problems in Topology II PDF Author: Elliott M. Pearl
Publisher: Elsevier
ISBN: 0080475299
Category : Mathematics
Languages : en
Pages : 777

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Book Description
This volume is a collection of surveys of research problems in topology and its applications. The topics covered include general topology, set-theoretic topology, continuum theory, topological algebra, dynamical systems, computational topology and functional analysis.* New surveys of research problems in topology* New perspectives on classic problems* Representative surveys of research groups from all around the world

Thin Sets in Discrete Abelian Groups

Thin Sets in Discrete Abelian Groups PDF Author: Ron C. Blei
Publisher:
ISBN:
Category : Dissertations, Academic
Languages : en
Pages : 128

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Mathematical Reviews

Mathematical Reviews PDF Author:
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 884

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Book Description