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E-raamat: Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis

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  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Mathematics 2160
  • Ilmumisaeg: 08-Jul-2016
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319397801
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Extensions of Positive Definite Functions: Applications and Their Harmonic AnalysisSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Mathematics 2160
  • Ilmumisaeg: 08-Jul-2016
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319397801
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This monograph deals with the mathematics of extending given partial data-sets obtained from experiments; Experimentalists frequently gather spectral data when the observed data is limited, e.g., by the precision of instruments; or by other limiting external factors. Here the limited information is a restriction, and the extensions take the form of full positive definite function on some prescribed group. It is therefore both an art and a science to produce solid conclusions from restricted or limited data.  While the theory of is important in many areas of pure and applied mathematics, it is difficult for students and for the novice to the field, to find accessible presentations which cover all relevant points of view, as well as stressing common ideas and interconnections. We have aimed at filling this gap, and we have stressed hands-on-examples.

Introduction.-Extensions of Continuous Positive Definite Functions.-The Case of More General Groups.-Examples.-Type I vs. Type II Extensions.-Spectral Theory for Mercer Operators, and Implications for Ext (F).-Green"s Functions.-Comparing the Different RKHSs HF and HK.-Comparing the Different RKHSs HF and HK.-Models for, and Spectral Representations of, Operator Extensions.-Overview and Open Questions.-List of Figures.-List of Tables.-References.-Index.
1 Introduction
1(16)
1.1 Two Extension Problems
3(2)
1.1.1 Where to Find It
3(2)
1.2 Quantum Physics
5(1)
1.3 Stochastic Processes
6(5)
1.3.1 Early Roots
6(4)
1.3.2 An Application of Lemma 1.1: A Positive Definite Function on an Infinite Dimensional Vector Space
10(1)
1.4 Overview of Applications of RKHSs
11(3)
1.4.1 Connections to Gaussian Processes
11(3)
1.5 Earlier Papers
14(1)
1.6 Organization
15(2)
2 Extensions of Continuous Positive Definite Functions
17(30)
2.1 The RKHS HF
18(5)
2.1.1 An Isometry
21(2)
2.2 The Skew-Hermitian Operator D(F) HF
23(11)
2.2.1 The Case of Conjugations
26(5)
2.2.2 Illustration: G = R, Correspondence Between the Two Extension Problems
31(3)
2.3 Enlarging the Hilbert Space
34(4)
2.4 Ext1(F) and Ext2(F)
38(5)
2.4.1 The Case of N = 1
40(1)
2.4.2 Comparison of p.d. Kernels
40(3)
2.5 Spectral Theory of D(F) and Its Extensions
43(4)
3 The Case of More General Groups
47(20)
3.1 Locally Compact Abelian Groups
47(7)
3.2 Lie Groups
54(13)
3.2.1 The GNS Construction
59(2)
3.2.2 Local Representations
61(2)
3.2.3 The Convex Operation in Ext (F)
63(4)
4 Examples
67(26)
4.1 The Case of G = Rn
67(5)
4.2 The Case of G = R/Z
72(3)
4.3 Example: ei2πx
75(2)
4.4 Example: e--|x| in (--a, a), Extensions to T = R/Z
77(4)
4.4.1 General Consideration
78(3)
4.5 Example: e--|x| in (--a, a), Extensions to R
81(8)
4.6 Example: A Non-extendable p.d. Function in a Neighborhood of Zero in G = R2
89(4)
4.6.1 A Locally Defined p.d. Functions F on G = R2 with Ext (F) = ø
91(2)
5 Type I vs. Type II Extensions
93(22)
5.1 Polya Extensions
93(6)
5.2 Main Theorems
99(7)
5.2.1 Some Applications
103(3)
5.3 The Deficiency-Indices of D(F)
106(4)
5.3.1 Polya-Extensions
108(2)
5.4 The Example 5.3, Green's Function, and an HF-ONB
110(5)
6 Spectral Theory for Mercer Operators, and Implications for Ext (F)
115(36)
6.1 Groups, Boundary Representations, and Renormalization
116(17)
6.2 Shannon Sampling, and Bessel Frames
133(4)
6.3 Application: The Case of F2 and Rank-1 Perturbations
137(5)
6.4 Positive Definite Functions, Green's Functions, and Boundary
142(9)
6.4.1 Connection to the Energy Form Hilbert Spaces
147(4)
7 Green's Functions
151(20)
7.1 The RKHSs for the Two Examples F2 and F3 in Table 5.1
151(18)
7.1.1 Green's Functions
152(3)
7.1.2 The Case of F2(x) = 1 -- |x|, x ε (--1/2, 1/2)
155(6)
7.1.3 The Case of F3(x) = e--|x|, x ε (--1, 1)
161(4)
7.1.4 Integral Kernels and Positive Definite Functions
165(1)
7.1.5 The Ornstein-Uhlenbeck Process Revisited
166(1)
7.1.6 An Overview of the Two Cases: F2 and F3
167(2)
7.2 Higher Dimensions
169(2)
8 Comparing the Different RKHSs F and K
171(22)
8.1 Applications
177(2)
8.2 Radially Symmetric Positive Definite Functions
179(2)
8.3 Connecting F and F When F Is a Positive Definite Function
181(2)
8.4 The Imaginary Part of a Positive Definite Function
183(10)
8.4.1 Connections to, and Applications of, Bochner's Theorem
187(6)
9 Convolution Products
193(4)
10 Models for, and Spectral Representations of, Operator Extensions
197(20)
10.1 Model for Restrictions of Continuous p.d. Functions on R
197(6)
10.2 A Model of ALL Deficiency Index-(1, 1) Operators
203(6)
10.2.1 Momentum Operators in L2 (0, 1)
207(2)
10.2.2 Restriction Operators
209(1)
10.3 The Case of Indices (d, d) Where d > 1
209(2)
10.4 Spectral Representation of Index (1, 1) Hermitian Operators
211(6)
11 Overview and Open Questions
217(2)
11.1 From Restriction Operator to Restriction of p.d. Function
217(1)
11.2 The Splitting HF = HF(atom) + HF(ac) + HF(sing)
217(1)
11.3 The Case of G = R1
218(1)
11.4 The Extreme Points of Ext (F) and {F}
218(1)
References 219(10)
Index 229
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