局部域上的调和分析与分形分析及其应用

PrefaceChapter 1 Preliminary1.1 Galois field GF （p）1.1.1 Galois field GF （p）， characteristic number p1.1.2 Algebraic extension fields of Galois field GF （p）1.2 Structures of local fields1.2.1 Definitions of local fields1.2.2 Valued structure of a local field Kq1.2.3 Haar measure and Harr integral on a local field Kq1.2.4 Important subsets in a local field Kq1.2.5 Base for neighborhood system of a local field Kq1.2.6 Expressions of elements in Kq， operations1.2.7 Important properties of balls in a local field Kp1.2.8 Order structures in a local field Kp1.2.9 Relationship between local field Kq and Euclidean space R ExercisesChapter 2 Character Group Fp of Local Field Kp2.1 Character groups of locally compact groups2.1.1 Characters of groups2.1.2 Characters and character groups of locally compact groups2.1.3 Pontryagin dual theorem 2.1.4 Examples2.2 Character group гp of Kp2.2.1 Properties of X ∈гp and гp2.2.2 Character group of p—series field Sp2.2.3 Character group of p—adic field Ap2.3 Some formulas in local fields2.3.1 Haar measures of certain important sets in Kp2.3.2 Integrals for characters in Kp2.3.3 Integrals for some functions in KpExercisesChapter 3 Harmonic Analysis on Local Fields3.1 Fourier analysis on a local field Kp3.1.1 L1—theory3.1.2 L2—theory3.1.3 Lr—theory 1＜r＜2 3.1.4 Distribution theory on KpExercises3.2 Pseudo—differential operators on local fields3.2.1 Symbol class Sαρδ（Kp）≡Sαρδ（Kp×гp）3.2.2 Pseudo—differential operator Tα on local fields3.3 p—type derivatives and p—type integrals on local fields3.3.1 p—type calculus on local fields3.3.2 Properties of p—type derivatives and p—type integrals of ψ∈S（Kp）3.3.3 p—type derivatives and p—type integrals of T∈S*（Kp）3.3.4 Background of establishing of p—type calculus3.4 Operator and construction theory of function on local fields3.4.1 Operators on a local field Kp3.4.2 Construction theory of function on a local field KpExercisesChapter 4 Function Spaces on Local Fields4.1 B—type spaces and F—type spaces on local fields4.1.1 B—type spaces， F—type spaces4.1.2 Special cases of B—type spaces and F—type spaces4.1.3 Holder type spaces on local fields4.1.4 Lebesgue type spaces and Sobolev type spacesExercises4.2 Lipschitz classes on local fields4.2.1 Lipschitz classes on local fields4.2.2 Chains of function spaces on Euclidean spaces4.2.3 The cases on local fields4.2.4 Comparison of Euclidean space analysis with local field analysisExercises4.3 Fractal spaces on local fields4.3.1 Fractal spaces on Kp4.3.2 Completeness of space K （（Kp），h） on Kp4.3.3 Some useful transformations on KpExercisesChapter 5 Fractal Analysis on Local Fields5.1 Fractal dimensions on local fields5.1.1 Hausdorff measure and dimension5.1.2 Box dimension5.1.3 Packing measure and dimensionExercises5.2 Analytic expressions of dimensions of sets in local fields5.2.1 Borel measure and Borel measurable sets5.2.2 distribution dimension5.2.3 Fourier dimensionExercises5.3 p—type calculus and fractal dimensions on local fields5.3.1 Structures of Kp， 3—adic Cantor type set， 3—adic Cantor type function5.3.2 p—type derivative and integral of r（x）on K3 5.3.3 p—type derivative and integral of Weierstrass type function on Kp5.3.4 p—type derivative and integral of second Weierstrass type function on KpExercisesChapter 6 Fractal PDE on Local Fields6.1 Special examples6.1.1 Classical 2—dimension wave equation with fractal boundary6.1.2 p—type 2—dimension wave equation with fractal boundary6.2 Further study on fractal analysis over local fields6.2.1 Pseudo—differential operator Tα 6.2.2 Further problems on fractal analysis over local fieldsExercisesChapter 7 Applications to Medicine Science7.1 Determine the malignancy of liver cancers7.1.1 Terrible havocs of liver cancer， solving idea7.1.2 The main methods in studying of liver cancers7.2 Examples in clinical medicine7.2.1 Take data from the materials ofliver cancers of patients7.2.2 Mathematical treatment for data7.2.3 Compute fractal dimensions7.2.4 Induce to obtain mathematical models7.2.5 Other problems in the research of liver cancersReferencesIndex