陶伯理论

所属分类:数学  
出版时间:2007-1   出版时间:科学分社   作者:科雷瓦   页数:483   字数:594000  

内容概要

陶伯理论对级数和积分的可求和性判定的不同方法加以比较,确定它们何时收敛,给出渐近估计和余项估计。由陶伯理论的最初起源开始,作者介绍该理论的发展历程:他的专业评论再现了早期结果所引来的兴奋;论及困难而令人着迷的哈代-李特尔伍德定理及其出人意料的一个简洁证明;高度赞扬维纳基于傅里叶理信论的突破,引人入胜的“高指数”定理以及应用于概率论的Karamata正则变分理论。作者也提及盖尔范德对维纳理论的代数处理以及基本人的分布方法。介绍了博雷尔方法和“圆”方法的一个统一的新理论,本书还讨论研究素数定理的各种陶伯方法。书后附有大量参考文献和详细尽的索引。

作者简介

作者:(荷兰)科雷瓦 (Jacob Korevaar)

书籍目录


The
Hardy-Littlewood
Theorems
1
Introduction
2
Examples
of
Summability
Methods
Abelian
Theorems
and
Tauberian
Question
3
Simple
Applications
of
Cesa(')ro,
Abel
and
Borel
Summability
4
Lambert
Summability
in
Number
Theory
5
Tauber's
Theorems
for
Abel
Summability
6
Tauberian
Theorem
for
Cesa(')ro
Summability
7
Hardy-Littlewood
Tauberians
for
Abel
Summability
8
Tauberians
Involving
Dirichlet
Series
9
Tauberians
for
Borel
Summability
10
Lambert
Tauberian
and
Prime
Number
Theorem
11
Karamata's
Method
for
Power
Series
12
Wielandt's
Variation
on
the
Method
13
Transition
from
Series
to
Integrals
14
Extension
of
Tauber's
Theorems
to
Laplace-Stieltjes
Transforms
15
Hardy-Littlewood
Type
Theorems
Involving
Laplace
Transforms
16
Other
Tauberian
Conditions:
Slowly
Decreasing
Functions
17
Asymptotics
for
Derivatives
18
Integral
Tauberians
for
Cesa(')ro
Summability
19
The
Method
of
the
Monotone
Minorant
20
Boundedness
Theorem
Involving
a
General-Kernel
Transform
21
Laplace-Stieltjes
and
Stieltjes
Transform
22
General
Dirichlet
Series
23
The
High-Indices
Theorem
24
Optimality
of
Tauberian
Conditions
25
Tauberian
Theorems
of
Nonstandard
Type
26
Important
Properties
of
the
Zeta
FunctionⅡ
Wiener's
Theory
1
Introduction
2
Wiener
Problem:
Pitt's
Form
3
Testing
Equation
for
Wiener
Kernels
4
Original
Wiener
Problem
5
Wiener's
Theorem
With
Additions
by
Pitt
6
Direct
Applications
of
the
Testing
Equations
7
Fourier
Analysis
of
Wiener
Kernels
8
The
Principal
Wiener
Theorems
9
Proof
of
the
Division
Theorem
10
Wiener
Families
of
Kernels
11
Distributional
Approach
to
Wiener
Theory
12
General
Tauberian
for
Lambert
SummabilitY
13
Wiener's
'Second
Tauberian
Theorem'
14
A
Wiener
Theorem
for
Series
15
Extensions
16
Discussion
of
the
Tauberian
Conditions
17
Landau-Ingham
Asymptotics
18
Ingham
Summability
19
Application
of
Wiener
Theory
to
Harmonic
FunctionsⅢ
Complex
Tauberian
Theorems
1
Introduction
2
A
Landau-Type
Tauberian
for
Dirichlet
Series
3
Mellin
Transforms
4
The
Wiener-Ikehara
Theorem
5
Newer
Approach
to
Wiener-Ikehara
6
Newman's
Way
to
the
PNT.
Work
of
Ingham
7
Laplace
Transforms
of
Bounded
Functions
8
Application
to
Dirichlet
Series
and
the
PNT
9
Laplace
Transforms
of
Functions
Bounded
From
Below
10
Tauberian
Conditions
Other
Than
Boundedness
11
An
Optimal
Constant
in
Theorem
10.1
12
Fatou
and
Riesz.
General
Dirichlet
Series
13
Newer
Extensions
of
Fatou-Riesz
14
Pseudofunction
Boundary
Behavior
15
Applications
to
Operator
Theory
16
Complex
Remainder
Theory
17
The
Remainder
in
Fatou's
Theorem
18
Remainders
in
Hardy-Littlewood
Theorems
Involving
Power
Series
19
A
Remainder
for
the
Stieltjes
TransformⅣ
Karamata's
Heritage:
Regular
Variation
1
Introduction
2
Slow
and
Regular
Variation
3
Proof
of
the
Basic
Properties
4
Possible
Pathology
5
Karamata's
Characterization
of
Regularly.
Varying
Functions
6
Related
Classes
of
Functions
7
Integral
Transforms
and
Regular
Variation:
Introduction
8
Karamata's
Theorem
for
Laplace
Transforms
9
Stieltjes
and
Other
Transforms
10
The
Ratio
Theorem
11
Beurling
Slow
Variation
12
A
Result
in
Higher-Order
Theory
13
Mercerian
Theorems
14
Proof
of
Theorem
13.2
15
Asymptotics
Involving
Large
Laplace
Transforms
16
Transforms
of
Exponential
Growth:
Logarithmic
Theory
17
Strong
Asymptotics:
General
Case
18
Application
to
Exponential
Growth
19
Very
Large
Laplace
Transforms
20
Logarithmic
Theory
for
Very
Large
Transforms
21
Large
Transforms:
Complex
Approach
22
Proof
of
Proposition
21.4
23
Asymptotics
for
Partitions
24
Two-Sided
Laplace
TransformsⅤ
Extensions
of
the
Classical
Theory
1
Introduction
2
Preliminaries
on
Banach
Algebras
3
Algebraic
Form
of
Wiener's
Theorem
4
Weighted
L1
Spaces
5
Gelfand's
Theory
of
Maximal
Ideals
6
Application
to
the
Banach
Algebra

=
(Lω,
C)
7
Regularity
Condition
for

8
The
Closed
Maximal
Ideals
in

9
Related
Questions
Involving
Weighted
Spaces
10
A
Boundedness
Theorem
of
Pitt
11
Proof
of
Theorem
10.2,
Part
1
12
Theorem
10.2:
Proof
that
S(y)
=
Q(eεY)
13
Theorem
10.2:
Proof
that
S(y)
=
Q{eφ(y)
14
Boundedness
Through
Functional
Analysis
15
Limitable
Sequences
as
Elements
of
an
FK-space
16
Perfect
Matrix
Methods
17
Methods
with
Sectional
Convergence
18
Existence
of
(Limitable)
Bounded
Divergent
Sequences
19
Bounded
Divergent
Sequences,
Continued
20
Gap
Tauberian
Theorems
21
The
Abel
Method
22
Recurrent
Events
23
The
Theorem
of
Erd6s,
Feller
and
Pollard
24
Milin's
Theorem
25
Some
Propositions
26
Proof
of
Milin's
TheoremⅥ
Borel
Summability
and
General
Circle
Methods
1
Introduction
2
The
Methods
B
and
B'
3
Borel
Summability
of
Power
Series
4
The
Borel
Polygon
5
General
Circle
Methods

6
Auxiliary
Estimates
7
Series
with
Ostrowski
Gaps
8
Boundedness
Results
9
Integral
Formulas
forLimitability
10
Integral
Formulas:
Case
of
Positive
Sn
11
First
Form
of
theTauberian
Theorem
12
General
Tauberian
Theorem
with
Schmidt's
Condition
13
Tauberian
Theorem:
Case
of
Positive
Sn
14
AnApplication
to
Number
Theory
15
High-Indices
Theorems
16
Restricted
High-Indices
Theorem
for
General
Circle
Methods
17
The
Borel
High-Indices
Theorem
18
Discussion
of
the
Tauberian
Conditions
19
Growth
of
Power
Series
with
Square-Root
Gaps
20
Euler
Summability
21
The
Taylor
Method
and
Other
Special
Circle
Methods
22
The
Special
Methods
as
Fλ-Methods
23
High-Indices
Theorems
for
Special
Methods
24
Power
Series
Methods
25
Proof
of
Theorem24.4Ⅶ
Tauberian
Remainder
Theory
1
Introduction
2
Power
Series
and
Laplace
Transforms:How
the
Theory
Developed
3
Theorems
for
Laplace
Transforms
4
Proof
of
Theorems
3.1
and
3.2
5
One-Sided
L
1
Approximation
6
Proof
of
Proposition
5.2
7
Approximation
of
Smooth
Functions
8
Proof
of
Approximation
Theorem
3.4
9
Vanishing
Remainders:
Theorem
3.3
10
Optimality
of
the
Remainder
Estimates
11
Dirichlet
Series
and
High
Indices
12
Proof
of
Theorem
11.2,
Continued
13
The
Fourier
Integral
Method:
Introduction
14
Fourier
Integral
Method:
A
Model
Theorem
15
Auxiliary
Inequality
of
Ganelius
16
Proof
of
the
Model
Theorem
17
A
More
General
Theorem
18
Application
to
Stieltjes
Transforms
19
Fourier
Integral
Method:
Laplace-Stieltjes
Transform
20
Related
Results
21
Nonlinear
Problems
of
Erd6s
for
Sequences
22
Introduction
to
the
Proof
of
Theorem
21.3
23
Proof
of
Theorem
21.3,
Continued
24
An
Example
and
Some
Remarks
25
Introduction
to
the
Proof
of
Theorem
21.5
26
The
Fundamental
Relation
and
a
Reduction
27
Proof
of
Theorem
25.1,
Continued
28
The
End
GameReferencesIndex

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