HARMONIC ANALYSIS AND FRACTAL ANALYSIS OVER LOCAL FIELDS ..

by Weiyi Su

Overview

This book is a monograph on harmonic analysis and fractal analysis over local fields. It can also be used as lecture notes/textbook or as recommended reading for courses on modern harmonic and fractal analysis. It is as reliable as Fourier Analysis on Local Fields published in 1975 which is regarded as the first monograph in this research field.The book is self-contained, with wide scope and deep knowledge, taking modern mathematics (such as modern algebra, point set topology, functional analysis, distribution theory, and so on) as bases. Specially, fractal analysis is studied in the viewpoint of local fields, and fractal calculus is established by pseudo-differential operators over local fields. A frame of fractal PDE is constructed based on fractal calculus instead of classical calculus. On the other hand, the author does his best to make those difficult concepts accessible to readers, illustrate clear comparison between harmonic analysis on Euclidean spaces and that on local fields, and at the same time provide motivations underlying the new concepts and techniques. Overall, it is a high quality, up to date and valuable book for interested readers.

Product Details

ISBN-13:	9789813200524
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	08/17/2017
Sold by:	Barnes & Noble
Format:	eBook
Pages:	332
File size:	42 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface v

1 Preliminary 1

1.1 Galois field GF(p) 1

1.1.1 Galois field GF(p), characteristic number p 1

1.1.2 Algebraic extension fields of Galois field GF(p) 3

1.2 Structures of local fields 5

1.2.1 Definitions of local fields 5

1.2.2 Valued structure of a local field K_q 6

1.2.3 Haar measure and Haar integral on a local field K_q 7

1.2.4 Important subsets in a local field K_q 9

1.2.5 Base for neighborhood system of a local field K_q 10

1.2.6 Expressions of elements in K_q and operations 11

1.2.7 Important properties of balls in a local field K_p 14

1.2.8 Order structure in a local field K_p 15

1.2.9 Relationship between local field K_p and Euclidean space R 17

Exercises 19

2 Character Group P_p of Local Field K_p 21

2.1 Character groups of locally compact groups 21

2.1.1 Characters of groups 21

2.1.2 Characters and character groups of locally compact groups 22

2.1.3 Pontryagin dual theorem 23

2.1.4 Examples 23

2.2 Character group Γ_p of K_p 25

2.2.1 Properties of X ∈ Γp and Γ_p 26

2.2.2 Character group of p-series field S_p 29

2.2.3 Character groups of p-adic field A_p 32

2.3 Some formulas in local fields 35

2.3.1 Haar measures of certain important sets in K_p 35

2.3.2 Integrals for characters in K_p 36

2.3.3 Integrals for some functions in K_p 38

Exercises 39

3 Harmonic Analysis on Local Fields 41

3.1 Fourier analysis on a local field K_p 41

3.1.1 L¹-theory 42

3.1.2 L²-theory 60

3.1.3 L^r-Theory, 1 < r &rt; 2 67

3.1.4 Distribution theory on K_p 69

Exercises 81

3.2 Pseudo-differential operators on local fields 82

3.2.1 Symbol class S^α_pδ(K_p) Ξ S^α_pδ (K_p x Γ) 82

3.2.2 Pseudo-differential operator T_σ on local fields 85

3.3 p-type derivatives and p-type integrals on local fields 88

3.3.1 p-type calculus on local fields 88

3.3.2 Properties of p-type derivatives and p-type integrals of π ∈ S(K_p) 89

3.3.3 p-type derivatives and p-type integrals of T ∈ S*(K_p) 92

3.3.4 Background of establishing for p-type calculus 94

3.4 Operator and construction theory of function on Local fields 102

3.4.1 Operators on a local field K_p 102

3.4.2 Construction theory of function on a local field K_p 105

Exercises 128

4 Function Spaces on Local Fields 129

4.1 B-type spaces and F-type spaces on local fields 129

4.1.1 B-type spaces, F-type spaces 129

4.1.2 Special cases of B-type spaces and F-type spaces 135

4.1.3 Hölder type spaces on local fields 136

4.1.4 Lebesgue type spaces and Sobolev type spaces 141

Exercises 147

4.2 Lipschitz class on local fields 148

4.2.1 Lipschitz classes on local fields 148

4.2.2 Chains of function spaces on Euclidean spaces 153

4.2.3 The cases on a local field K_p 157

4.2.4 Comparison of Euclidean space analysis and local field analysis 159

Exercises 162

4.3 Fractal spaces on local feilds 162

4.3.1 Fractal spaces on K_p 163

4.3.2 Completeness of (K(K_p), h) on K_p 164

4.3.3 Some useful transformations on K_p 172

Exercises 182

5 Fractal Analysis on Local Fields 183

5.1 Fractal dimensions on local fields 183

5.1.1 Hausdorff measure and dimension 183

5.1.2 Box dimension 190

5.1.3 Packing measure and dimension 196

Exercises 200

5.2 Analytic expressions of dimensions of sets in local fields 200

5.2.1 Borel measure and Borel measurable sets 200

5.2.2 Distribution dimension 201

5.2.3 Fourier dimension 210

Exercises 213

5.3 p-type calculus and fractal dimensions on local fields 213

5.3.1 Structures of K_p, 3-adic Cantor type set, 3-adic Cantor type function 213

5.3.2 p-type derivative and p-type integral of V(x) K₃ 218

5.3.3 p-type derivative and integral of Weierstrass type function on K_p 226

5.3.4 p-type derivative and integral of second Weierstrass type function on K_p 233

Exercises 242

6 Fractal PDE on Local Fields 243

6.1 Special examples 244

6.1.1 Classical 2-dimension wave equation with fractal boundary 244

6.1.2 p-type 2-dimension wave equation with fractal boundary 255

6.2 Further study on fractal analysis over local fields 266

6.2.1 Pseudo-differential operator T_α 266

6.2.2 Further problems on fractal analysis over local fields 281

Exercises 282

7 Applications to Medicine Science 283

7.1 Determine the malignancy of liver cancers 284

7.1.1 Terrible havocs of liver cancer, solving idea 284

7.1.2 The main methods in studying of liver cancers 287

7.2 Examples in clinical medicine 291

7.2.1 Take data from the materials of liver cancers of patients 291

7.2.2 Mathematical treatment for data 291

7.2.3 Compute fractal dimensions 300

7.2.4 Induce to obtain mathematical models 303

7.2.5 Other problems in the research of liver cancers 304

Bibliography 305

Index 315