## An Introduction to Sato's HyperfunctionsThis book is a translation, with corrections and an updated bibliography, of Morimoto's 1976 book on the theory of hyperfunctions originally written in Japanese. Since the time that Sato established the theory of hyperfunctions, there have been many important applications to such areas as pseudodifferential operators and S-matrices. Assuming as little background as possible on the part of the reader, Morimoto covers the basic notions of the theory, from hyperfunctions of one variable to Sato's fundamental theorem. This book provides an excellent introduction to this important field of research. |

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### Contents

1 | |

3 | |

3 Power series and Reinhardt domains | 6 |

4 Linear topological spaces of holomorphic functions | 8 |

5 Germs of holomorphic functions | 11 |

K Runge open sets | 13 |

7 The FourierBorel transformation | 14 |

8 Entire functions of exponential type | 16 |

3 Various operations | 47 |

4 Jfunction and Kfunction | 49 |

5 Power functions | 53 |

6 Singular spectrum | 55 |

7 Relation with local analytic functionals | 58 |

8 Ordinary differential equations | 61 |

9 Distributions and hyperfunctions | 64 |

Cohomology Groups with Coefficients in a Sheaf | 69 |

Analytic Functional of One Variable | 23 |

1 The CauchyHilbert transformation | 24 |

2 The Runge theorems | 30 |

3 The MittagLeffler theorem | 32 |

4 A representation of analytic functional | 35 |

The FourierLaplace transform of an entire function of exponential type | 36 |

K Convolution | 38 |

Hyperfunctions of One Variable | 41 |

Definition of hyperfunctions | 43 |

2 Locality of hyperfunctions | 45 |

Cohomology Groups with Coefficients in f109 | 109 |

Analytic Functionals of Several Variables | 129 |

Hyperfunctions of Several Variables | 155 |

Microfunctions | 177 |

Development of Hyperfunction Theory | 211 |

Appendix A Linear Topological Spaces | 239 |

Appendix B Rudiments of Homological Algebra | 257 |

Bibliography | 265 |

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### Common terms and phrases

Abelian groups algebra assume Banach spaces called canonical mapping Chapter closed set cohomology groups compact set compact support complex neighborhood consider converges convex set Corollary Definition denote dual space Edge-of-the-Wedge theorem Example family of supports flabby resolution flabby sheaf following theorem Fourier-Borel transformation FS space Hausdorff holomorphic functions hyperfunction implies inductive limit Lemma linear space linear topological space locally closed set locally convex space long exact sequence micro-analytic microfunctions notation open covering open neighborhood open set presheaf proof of Theorem properly convex Proposition prove real analytic functions relative cohomology groups resp restriction mapping S. S. g satisfies the condition sequence of sheaves sheaf 9 sheaf homomorphism sheaf of germs special polyhedron Stein open set subset supp Suppose surjective system of seminorms theory of hyperfunctions topological space topology vanishes

### Popular passages

Page 1 - The main purpose of this chapter is to familiarize the reader with the notation, the definitions, and the theorems which will be used . frequently throughout this book.