LEADER 07488nam 22008055 450 001 996466771603316 005 20200703101421.0 010 $a3-319-39780-X 024 7 $a10.1007/978-3-319-39780-1 035 $a(CKB)3710000000765391 035 $a(DE-He213)978-3-319-39780-1 035 $a(MiAaPQ)EBC6301229 035 $a(MiAaPQ)EBC5595835 035 $a(Au-PeEL)EBL5595835 035 $a(OCoLC)953574514 035 $a(PPN)194511731 035 $a(EXLCZ)993710000000765391 100 $a20160708d2016 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aExtensions of Positive Definite Functions$b[electronic resource] $eApplications and Their Harmonic Analysis /$fby Palle Jorgensen, Steen Pedersen, Feng Tian 205 $a1st ed. 2016. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2016. 215 $a1 online resource (XXVI, 231 p. 48 illus., 9 illus. in color.) 225 1 $aLecture Notes in Mathematics,$x0075-8434 ;$v2160 311 $a3-319-39779-6 327 $aIntro -- Foreword -- Preface -- Acknowledgments -- Contents -- List of Figures -- List of Tables -- Symbols -- 1 Introduction -- 1.1 Two Extension Problems -- 1.1.1 Where to Find It -- 1.2 Quantum Physics -- 1.3 Stochastic Processes -- 1.3.1 Early Roots -- 1.3.2 An Application of Lemma 1.1: A Positive Definite Function on an Infinite Dimensional Vector Space -- 1.4 Overview of Applications of RKHSs -- 1.4.1 Connections to Gaussian Processes -- 1.5 Earlier Papers -- 1.6 Organization -- 2 Extensions of Continuous Positive Definite Functions -- 2.1 The RKHS HF -- 2.1.1 An Isometry -- 2.2 The Skew-Hermitian Operator D(F) in HF -- 2.2.1 The Case of Conjugations -- 2.2.2 Illustration: G=R, Correspondence Between the Two Extension Problems -- 2.3 Enlarging the Hilbert Space -- 2.4 Ext1(F) and Ext2(F) -- 2.4.1 The Case of n=1 -- 2.4.2 Comparison of p.d. Kernels -- 2.5 Spectral Theory of D(F) and Its Extensions -- 3 The Case of More General Groups -- 3.1 Locally Compact Abelian Groups -- 3.2 Lie Groups -- 3.2.1 The GNS Construction -- 3.2.2 Local Representations -- 3.2.3 The Convex Operation in Ext(F) -- 4 Examples -- 4.1 The Case of G=Rn -- 4.2 The Case of G=R/Z -- 4.3 Example: ei2?x -- 4.4 Example: e-|x| in (-a,a), Extensions to T=R/Z -- 4.4.1 General Consideration -- 4.5 Example: e-|x| in (-a,a), Extensions to R -- 4.6 Example: A Non-extendable p.d. Function in a Neighborhood of Zero in G=R2 -- 4.6.1 A Locally Defined p.d. Functions F on G=R2 with Ext(F)=. -- 5 Type I vs. Type II Extensions -- 5.1 Po?lya Extensions -- 5.2 Main Theorems -- 5.2.1 Some Applications -- 5.3 The Deficiency-Indices of D(F) -- 5.3.1 Po?lya-Extensions -- 5.4 The Example 5.3, Green's Function, and an HF-ONB -- 6 Spectral Theory for Mercer Operators, and Implications for Ext(F) -- 6.1 Groups, Boundary Representations, and Renormalization -- 6.2 Shannon Sampling, and Bessel Frames. 327 $a6.3 Application: The Case of F2 and Rank-1 Perturbations -- 6.4 Positive Definite Functions, Green's Functions, and Boundary -- 6.4.1 Connection to the Energy Form Hilbert Spaces -- 7 Green's Functions -- 7.1 The RKHSs for the Two Examples F2 and F3 in Table 5.1 -- 7.1.1 Green's Functions -- 7.1.1.1 Summary: Conclusions for the Two Examples -- 7.1.2 The Case of F2(x)=1-|x|, x(-12,12) -- 7.1.2.1 Pinned Brownian Motion -- 7.1.3 The Case of F3(x)=e-|x|, x(-1,1) -- 7.1.4 Integral Kernels and Positive Definite Functions -- 7.1.5 The Ornstein-Uhlenbeck Process Revisited -- 7.1.6 An Overview of the Two Cases: F2 and F3. -- 7.2 Higher Dimensions -- 8 Comparing the Different RKHSs HF and HK -- 8.1 Applications -- 8.2 Radially Symmetric Positive Definite Functions -- 8.3 Connecting F and F When F Is a Positive Definite Function -- 8.4 The Imaginary Part of a Positive Definite Function -- 8.4.1 Connections to, and Applications of, Bochner's Theorem -- 9 Convolution Products -- 10 Models for, and Spectral Representations of, OperatorExtensions -- 10.1 Model for Restrictions of Continuous p.d. Functions on R -- 10.2 A Model of ALL Deficiency Index-(1,1) Operators -- 10.2.1 Momentum Operators in L2(0,1) -- 10.2.2 Restriction Operators -- 10.3 The Case of Indices (d,d) Where d>1 -- 10.4 Spectral Representation of Index (1,1) Hermitian Operators -- 11 Overview and Open Questions -- 11.1 From Restriction Operator to Restriction of p.d. Function -- 11.2 The Splitting HF=HF(atom)HF(ac) HF(sing) -- 11.3 The Case of G=R1 -- 11.4 The Extreme Points of Ext(F) and { F} -- References -- Index. 330 $aThis monograph deals with the mathematics of extending given partial data-sets obtained from experiments; Experimentalists frequently gather spectral data when the observed data is limited, e.g., by the precision of instruments; or by other limiting external factors. Here the limited information is a restriction, and the extensions take the form of full positive definite function on some prescribed group. It is therefore both an art and a science to produce solid conclusions from restricted or limited data. While the theory of is important in many areas of pure and applied mathematics, it is difficult for students and for the novice to the field, to find accessible presentations which cover all relevant points of view, as well as stressing common ideas and interconnections. We have aimed at filling this gap, and we have stressed hands-on-examples. 410 0$aLecture Notes in Mathematics,$x0075-8434 ;$v2160 606 $aHarmonic analysis 606 $aTopological groups 606 $aLie groups 606 $aFourier analysis 606 $aFunctional analysis 606 $aMathematical physics 606 $aProbabilities 606 $aAbstract Harmonic Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12015 606 $aTopological Groups, Lie Groups$3https://scigraph.springernature.com/ontologies/product-market-codes/M11132 606 $aFourier Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12058 606 $aFunctional Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12066 606 $aMathematical Physics$3https://scigraph.springernature.com/ontologies/product-market-codes/M35000 606 $aProbability Theory and Stochastic Processes$3https://scigraph.springernature.com/ontologies/product-market-codes/M27004 615 0$aHarmonic analysis. 615 0$aTopological groups. 615 0$aLie groups. 615 0$aFourier analysis. 615 0$aFunctional analysis. 615 0$aMathematical physics. 615 0$aProbabilities. 615 14$aAbstract Harmonic Analysis. 615 24$aTopological Groups, Lie Groups. 615 24$aFourier Analysis. 615 24$aFunctional Analysis. 615 24$aMathematical Physics. 615 24$aProbability Theory and Stochastic Processes. 676 $a515.2433 700 $aJorgensen$b Palle$4aut$4http://id.loc.gov/vocabulary/relators/aut$0785653 702 $aPedersen$b Steen$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aTian$b Feng$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996466771603316 996 $aExtensions of Positive Definite Functions$92004330 997 $aUNISA