## Extensions of Positive Definite Functions: Applications and Their Harmonic AnalysisThis 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. |

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### Contents

1 | |

2 Extensions of Continuous Positive Definite Functions | 17 |

3 The Case of More General Groups | 47 |

4 Examples | 67 |

5 Type I vs Type II Extensions | 93 |

6 Spectral Theory for Mercer Operators and Implications for ExtF | 115 |

7 Greens Functions | 151 |

8 Comparing the Different RKHSs HF and HK | 170 |

9 Convolution Products | 193 |

10 Models for and Spectral Representations of OperatorExtensions | 196 |

11 Overview and Open Questions | 217 |

219 | |

229 | |

### Common terms and phrases

ˇ ˇ ˇ Abelian groups applications assume Bochner Borel measure boundary Brownian motion compute continuous functions continuous p.d. function continuous positive definite Corollary deficiency indices 1;1 defined p.d. function denotes dense domain Example Ext F Ext.F extension of F extension problem Extensions of Positive fixed follows function F G D Rn Gaussian group G Hermitian operator Hilbert space inner product isometry kernel Hilbert space Lebesgue measure Lemma Let F Lie algebra Lie groups locally compact Abelian locally defined p.d. Math Mercer operator operator TF operators with dense P.E.T. Jorgensen positive definite functions positive definite p.d. probability measure Proof Remark reproducing kernel Hilbert RKHS RKHS HF RKHSs Sect selfadjoint selfadjoint extensions selfadjoint operator skew-adjoint extension skew-Hermitian operator ſº spectral theory spline extension stochastic processes subset subspace Theorem unbounded unitary representation vector