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El. knyga: Harmonic Analysis And Fractal Analysis Over Local Fields And Applications

(Nanjing Univ, China)
  • Formatas: 332 pages
  • Išleidimo metai: 17-Aug-2017
  • Leidėjas: World Scientific Publishing Co Pte Ltd
  • Kalba: eng
  • ISBN-13: 9789813200524
Kitos knygos pagal šią temą:
Harmonic Analysis And Fractal Analysis Over Local Fields And ApplicationsDidesnis paveikslėlis
  • Formatas: 332 pages
  • Išleidimo metai: 17-Aug-2017
  • Leidėjas: World Scientific Publishing Co Pte Ltd
  • Kalba: eng
  • ISBN-13: 9789813200524
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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.
Preface v
1 Preliminary
1(20)
1.1 Galois field GF(p)
1(4)
1.1.1 Galois field GF(p), characteristic number p
1(2)
1.1.2 Algebraic extension fields of Galois field GF(p)
3(2)
1.2 Structures of local fields
5(16)
1.2.1 Definitions of local fields
5(1)
1.2.2 Valued structure of a local field Kq
6(1)
1.2.3 Haar measure and Haar integral on a local field Kq
7(2)
1.2.4 Important subsets in a local field Kq
9(1)
1.2.5 Base for neighborhood system of a local field Kq
10(1)
1.2.6 Expressions of elements in Kq and operations
11(3)
1.2.7 Important properties of balls in a local field Kp
14(1)
1.2.8 Order structure in a local field Kv
15(2)
1.2.9 Relationship between local field Kp and Euclidean space R
17(2)
Exercises
19(2)
2 Character Group Tp of Local Field Kp
21(20)
2.1 Character groups of locally compact groups
21(4)
2.1.1 Characters of groups
21(1)
2.1.2 Characters and character groups of locally compact groups
22(1)
2.1.3 Pontryagin dual theorem
23(1)
2.1.4 Examples
23(2)
2.2 Character group Γp of Kp
25(10)
2.2.1 Properties of χ ε Γp and Γp
26(3)
2.2.2 Character group of p-series field Sp
29(3)
2.2.3 Character groups of p-adic field Ap
32(3)
2.3 Some formulas in local fields
35(6)
2.3.1 Haar measures of certain important sets in Kp
35(1)
2.3.2 Integrals for characters in Kp
36(2)
2.3.3 Integrals for some functions in Kp
38(1)
Exercises
39(2)
3 Harmonic Analysis on Local Fields
41(88)
3.1 Fourier analysis on a local field Kp
41(41)
3.1.1 L1-theory
42(18)
3.1.2 L2-theory
60(7)
3.1.3 Lr-Theory, 1 < r < 2
67(2)
3.1.4 Distribution theory on Kp
69(12)
Exercises
81(1)
3.2 Pseudo-differential operators on local fields
82(6)
3.2.1 Symbol class Sαps(Kp) = Sαps (Kp × Γp)
82(3)
3.2.2 Pseudo-differential operator Tσ on local fields
85(3)
3.3 p-type derivatives and p-type integrals on local fields
88(14)
3.3.1 p-type calculus on local fields
88(1)
3.3.2 Properties of p-type derivatives and p-type integrals of φ ε S(KP)
89(3)
3.3.3 p-type derivatives and p-type integrals of T ε S*(Kp)
92(2)
3.3.4 Background of establishing for p-type calculus
94(8)
3.4 Operator and construction theory of function on Local fields
102(27)
3.4.1 Operators on a local field Kp
102(3)
3.4.2 Construction theory of function on a local field Kp
105(23)
Exercises
128(1)
4 Function Spaces on Local Fields
129(54)
4.1 B-type spaces and F-type spaces on local fields
129(19)
4.1.1 B-type spaces, F-type spaces
129(6)
4.1.2 Special cases of B-type spaces and F-type spaces
135(1)
4.1.3 Holder type spaces on local fields
136(5)
4.1.4 Lebesgue type spaces and Sobolev type spaces
141(6)
Exercises
147(1)
4.2 Lipschitz class on local fields
148(14)
4.2.1 Lipschitz classes on local fields
148(5)
4.2.2 Chains of function spaces on Euclidean spaces
153(4)
4.2.3 The cases on a local field Kp
157(2)
4.2.4 Comparison of Euclidean space analysis and local field analysis
159(3)
Exercises
162(1)
4.3 Fractal spaces on local feilds
162(21)
4.3.1 Fractal spaces on Kp
163(1)
4.3.2 Completeness of (K(Kp),h) on Kp
164(8)
4.3.3 Some useful transformations on Kp
172(10)
Exercises
182(1)
5 Fractal Analysis on Local Fields
183(60)
5.1 Fractal dimensions on local fields
183(17)
5.1.1 Hausdorff measure and dimension
183(7)
5.1.2 Box dimension
190(6)
5.1.3 Packing measure and dimension
196(4)
Exercises
200(1)
5.2 Analytic expressions of dimensions of sets in local fields
200(13)
5.2.1 Borel measure and Borel measurable sets
200(1)
5.2.2 Distribution dimension
201(9)
5.2.3 Fourier dimension
210(3)
Exercises
213(1)
5.3 p-type calculus and fractal dimensions on local fields
213(30)
5.3.1 Structures of Kp, 3-adic Cantor type set, 3-adic Cantor type function
213(5)
5.3.2 p-type derivative and p-type integral of v(x) on K3
218(8)
5.3.3 p-type derivative and integral of Weierstrass type function on Kp
226(7)
5.3.4 p-type derivative and integral of second Weierstrass type function on Kp
233(9)
Exercises
242(1)
6 Fractal PDE on Local Fields
243(40)
6.1 Special examples
244(22)
6.1.1 Classical 2-dimension wave equation with fractal boundary
244(11)
6.1.2 p-type 2-dimension wave equation with fractal boundary
255(11)
6.2 Further study on fractal analysis over local fields
266(17)
6.2.1 Pseudo-differential operator Tα
266(15)
6.2.2 Further problems on fractal analysis over local fields
281(1)
Exercises
282(1)
7 Applications to Medicine Science
283(22)
7.1 Determine the malignancy of liver cancers
284(7)
7.1.1 Terrible havocs of liver cancer, solving idea
284(3)
7.1.2 The main methods in studying of liver cancers
287(4)
7.2 Examples in clinical medicine
291(14)
7.2.1 Take data from the materials of liver cancers of patients
291(1)
7.2.2 Mathematical treatment for data
291(9)
7.2.3 Compute fractal dimensions
300(3)
7.2.4 Induce to obtain mathematical models
303(1)
7.2.5 Other problems in the research of liver cancers
304(1)
Bibliography 305(10)
Index 315
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