Number theory, Fourier anaylysis and geometric discrepancy:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
2014
|
| Edition: | 1. publ. |
| Series: | London Mathematical Society student texts
81 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | X, 240 S. graph. Darst. |
| ISBN: | 9781107044036 9781107619852 |
Staff View
MARC
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| 100 | 1 | |a Travaglini, Giancarlo |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Number theory, Fourier anaylysis and geometric discrepancy |c Giancarlo Travaglini |
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Record in the Search Index
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| adam_text | Contents
Introduction page
ix
PART
ONE ELEMENTARY NUMBER THEORY
1
Prelude
3
1.1
Prime numbers and factorization
3
1.2
Chebyshev s theorem
11
Arithmetic functions and integer points
19
2.1
Arithmetic functions and Dirichlet product
19
2.2
μ(η)
23
2.3
d(n) and
crin)
25
2.4
r
(η)
33
2.5
φ(η)
35
Congruences
41
3.1
Basic properties
42
3.2
The theorems of
Fermat
and
Euler
46
3.3
Almost prime numbers
50
3.4
The Chinese remainder theorem
56
3.5
The RSA encryption
59
Quadratic reciprocity and Fourier series
65
4.1
Quadratic residues
66
4.2
Gauss sums
74
4.3
Fourier series
76
4.4
Another proof of the quadratic reciprocity law
83
Sums of squares
87
5.1
The theorems of Minkowski and Dirichlet
87
vn
viii Contents
5.2
Sums of two
squares
92
5.3
Gaussian integers
94
5.4
Sums of four squares
100
PART TWO FOURIER ANALYSIS AND GEOMETRIC
DISCREPANCY
105
6
Uniform distribution and completeness of the trigonometric
system
107
6.1
Kronecker s theorem
108
6.2
Completeness of the trigonometric system
110
6.3
The Weyl criterion
118
6.4
Normal numbers
123
6.5
Benford s
law
127
7
Discrepancy and trigonometric approximation
134
7.1
One-sided trigonometric approximation
135
7.2
The
Erdős-Turán
inequality
138
7.3
The inequalities of Koksma and Koksma-Hlawka
140
8
Integer points and
Poisson
summation formula
150
8.1
Fourier integrals
150
8.2
The
Poisson
summation formula
160
8.3
The Gauss circle problem
162
8.4
Integer points in convex bodies
169
9
Integer points and exponential sums
183
9.1
Preliminary results
183
9.2
Van
der Corpuťs
method and the Dirichlet divisor
problem
186
10
Geometric discrepancy and decay of Fourier transforms
193
10.1
Choosing
N
points in a cube
193
10.2
Roth s theorem
195
10.3
Average decay of Fourier transforms
200
10.4
Irregularities of distribution for convex bodies
203
11
Discrepancy in high dimension and Bessel functions
214
11.1
Bessel functions
215
11.2
Deterministic and probabilistic discrepancies
2І9
11.3
The cased-
5,9, 13,... 224
References
229
Index
238
The study of geometric discrepancy, which provides a framework for quanti-
fying the quality of a distribution of a finite set of points, has experienced
significant growth in recent decades. This book provides a self-contained
course in number theory, Fourier analysis and geometric discrepancy theory,
and the relations between them, at the advanced undergraduate or beginning
graduate level. It starts as a traditional course in elementary number theory, and
introduces the reader to subsequent material on uniform distribution of infinite
sequences and discrepancy of finite sequences. Both modern and classical
aspects of the theory are discussed, such as Weyl s criterion, Bedford’s law, the
Koksma-Hlawka inequality, lattice point problems, and irregularities of distri-
bution for convex bodies. Fourier analysis also features prominently, for which
the theory is developed in parallel, including topics such as convergence of
Fourier series, one-sided trigonometric approximation, the Poisson summation
formula, exponential sums, decay of Fourier transforms, and Bessel functions.
|
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| author_facet | Travaglini, Giancarlo |
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| dewey-full | 512.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.7 |
| dewey-search | 512.7 |
| dewey-sort | 3512.7 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV042073684 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:12:00Z |
| institution | BVB |
| isbn | 9781107044036 9781107619852 |
| language | English |
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| physical | X, 240 S. graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | London Mathematical Society student texts |
| series2 | London Mathematical Society student texts |
| spelling | Travaglini, Giancarlo Verfasser aut Number theory, Fourier anaylysis and geometric discrepancy Giancarlo Travaglini 1. publ. Cambridge Cambridge Univ. Press 2014 X, 240 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society student texts 81 Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s DE-604 London Mathematical Society student texts 81 (DE-604)BV000841726 81 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027514895&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027514895&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Travaglini, Giancarlo Number theory, Fourier anaylysis and geometric discrepancy London Mathematical Society student texts Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4067277-3 |
| title | Number theory, Fourier anaylysis and geometric discrepancy |
| title_auth | Number theory, Fourier anaylysis and geometric discrepancy |
| title_exact_search | Number theory, Fourier anaylysis and geometric discrepancy |
| title_full | Number theory, Fourier anaylysis and geometric discrepancy Giancarlo Travaglini |
| title_fullStr | Number theory, Fourier anaylysis and geometric discrepancy Giancarlo Travaglini |
| title_full_unstemmed | Number theory, Fourier anaylysis and geometric discrepancy Giancarlo Travaglini |
| title_short | Number theory, Fourier anaylysis and geometric discrepancy |
| title_sort | number theory fourier anaylysis and geometric discrepancy |
| topic | Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Zahlentheorie |
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