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商品コード: 9781680835922

Higher-order Fourier Analysis and Applications

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書名

Higher-order Fourier Analysis and Applications
シリーズ名 Foundations and Trends in Theoretical Computer Science (Vol. 13, Issue 4)
著者・編者 Hatami, H. et al.
出版社/発行元 Now publishers
発行年/月 2019年9月   
装丁 電子ブック
ページ数/巻数

230 ページ

ISBN 978-1-68083-592-2

 


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Description

 

Higher-order Fourier Analysis and Applications provides an introduction to the field of higher-order Fourier analysis with an emphasis on its applications to theoretical computer science. Higher-order Fourier analysis is an extension of the classical Fourier analysis. It has been developed by several mathematicians over the past few decades in order to study problems in an area of mathematics called additive combinatorics, which is primarily concerned with linear patterns such as arithmetic progressions in subsets of integers.

The monograph is divided into three parts: Part I discusses linearity testing and its generalization to higher degree polynomials. Part II present the fundamental results of the theory of higher-order Fourier analysis. Part III uses the tools developed in Part II to prove some general results about property testing for algebraic properties. It describes applications of the theory of higher-order Fourier analysis in theoretical computer science, and, to this end, presents the foundations of this theory through such applications; in particular to the area of property testing.

 

Contents:

 

1. Introduction

Part I. Low Degree Testing
2. Low Degree Testing
3. Low-degree Tests, the 99% Regime
4. Low-degree Tests, the 1% Regime
5. Gowers Norms, the Inverse Gowers Conjecture and its Failure

Part II. Higher Order Fourier Analysis

6. Nonclassical Polynomials, and the Inverse Gowers Theorem
7. Rank, Regularity, and Other Notions of Uniformity
8. Bias vs Low Rank in Large Fields
9. Decomposition Theorems
10. Homogeneous Nonclassical Polynomials
11. Complexity of Systems of Linear Forms
12. Deferred Technical Proofs
13. Algorithmic Regularity

Part III. Algebraic Property Testing
14. Algebraic Properties
15. One-Sided Algebraic Property Testing
16. Degree Structural Properties
17. Estimating the Distance from Algebraic Properties

Part IV. Open Problems
18. Open Problems

References

 

 

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