## Distributions and Convolution EquationsTo make their work more accessible to readers new to this field, the authors restrict initial treatment of problems to the half-line and formulate only principal results, in their simplest form. Special results and possible generalizations are presented as problems and exercises. For this reason this work is recommended not only for experts in the field of partial differential equations, but also for senior undergraduate and graduate students less familiar with this area. The authors apply the results of many years of their own original research to a systematic presentation of the theory of distributions. The first part of their monograph is devoted to the Cauchy problem. The authors show that Petrovskii's classic theory of the correctness of the Cauchy problem for general differential operators with constant convolution equations in certain spaces of distributions. The second part deals with the Wiener-Hopf equation and related topics in the theory of boundary value problems for convolution equa |

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### Contents

1 Spaces of Testfunctions and Distributions | 3 |

2 The Scale of Hilbert Spaces Associated with S Sobolev | 18 |

Appendix to 2 | 31 |

3 Convolution in Spaces of Tempered Distributions | 37 |

Appendix to 3 | 48 |

5 The Spaces Sjoy and the Related Scales | 60 |

Appendix to Chapter 1 Scales of Topological Linear Spaces | 73 |

Homogeneous Cauchy Problem for Convolution | 93 |

4 Exponentially Correct Differential Operators | 240 |

Nonhomogeneous Cauchy Problem for Convolution | 249 |

Appendix to 1 | 274 |

Cauchy Problem for Differential and Pseudodifferential | 311 |

Appendix to şi | 342 |

3 Pseudodifferential Equations in R the Case of the Spaces | 368 |

4 Exponentially Correct Differential Operators with Variable | 378 |

WienerHopf Convolutors and Boundary Value | 383 |

Appendix to şi | 113 |

3 Scales of Spaces of Functions Defined in R | 123 |

4 Convolution Operators and Convolution Equations in Spaces | 145 |

Convolutors and Convolution Equations in Spaces | 173 |

Convolution Equations in Spaces of Exponentially | 193 |

The Homogeneous Cauchy Problem in Spaces | 218 |

Appendix to 2 | 232 |

2 Factorization of WienerHopf Convolutors | 393 |

Appendix to 2 | 402 |

The WienerHopf Equation on a Halfline | 410 |

4 The WienerHopf Equation in a Halfspace | 430 |

453 | |

463 | |

### Common terms and phrases

According algebra analogous Appendix Applying arbitrary assertion assume Banach spaces belonging bounded canonical Chapter coefficients coincides commutable condition consequently consider consists constant construct continuous operator convergent convolution operator convolutors correct corresponding decreasing defined definition denote depend derive determined differential operators distributions element embeddings equivalent estimate exists exponentially extended fact factor spaces finite follows formula Fourier transform functions Further Hence holds holomorphic implies inclusion inductive limit inequality integral introduce inverse operator isomorphism Lemma linear mapping means natural necessary neighborhood norm obtain polynomial possesses present problem proof Proposition proved regular relation relative remains Remark replaced represented respect restriction right-hand side satisfying scale Section sense smooth solution solvability subspace sufficient symbols Theorem theory topology translations unique variables virtue whence zero