## Pick Interpolation and Hilbert Function SpacesThe book first rigorously develops the theory of reproducing kernel Hilbert spaces. The authors then discuss the Pick problem of finding the function of smallest $Hinfty$ norm that has specified values at a finite number of points in the disk. Their viewpoint is to consider $Hinfty$ as the multiplier algebra of the Hardy space and to use Hilbert space techniques to solve the problem. This approach generalizes to a wide collection of spaces. The authors then consider theinterpolation problem in the space of bounded analytic functions on the bidisk and give a complete description of the solution. They then consider very general interpolation problems. The book includes developments of all the theory that is needed, including operator model theory, the Arveson extension theorem,and the hereditary functional calculus. |

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### Contents

Prerequisites and Notation i | 1 |

Introduction | 7 |

Kernels and Function Spaces | 15 |

Hardy Spaces | 35 |

P2i | 49 |

Pick Redux | 55 |

Qualitative Properties of the Solution of the Pick Problem | 71 |

Characterizing Kernels with the Complete Pick Property | 79 |

Isometries | 151 |

The Bidisk | 167 |

The Extremal Three Point Problem on D2 | 195 |

Collections of Kernels | 211 |

Function Spaces | 237 |

Localization | 263 |

Appendix A Schur Products | 273 |

The Spectral Theorem for Normal mTuples | 287 |

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### Common terms and phrases

adjoint admissible kernel analytic assume Banach Bergman space bidisk Blaschke product bounded point evaluations C*-algebra Carleson Carleson measure Chapter choose closed unit ball co-isometric extension commutant lifting theorem complete Pick kernel complete Pick property completely positive complex numbers constant coordinate function Corollary define Definition denote dilation direct sum Dirichlet space disk Exercise finite set function f functional calculus Hardy space hence Hilbert function space Hilbert space holomorphic functions holomorphic space inequality interpolating sequence interpolation invariant subspace isometry kernel function Lebesgue measure Lemma Let H m-tuple matrix-valued Moreover multiplier algebra nodes non-zero normalized notation orthogonal projection orthonormal basis pair Pick matrix Pick problem positive semi-definite proof of Theorem prove rank rational function realization formula representation satisfies says Schur product Section self-adjoint space H span spectral subset Suppose Szego kernel uniform algebra unitary values vanish vector weak kernels weak-star topology weakly separated zero set