## Sobolev Spaces of Infinite Order and Differential Equations |

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### Contents

I | 11 |

II | 12 |

III | 18 |

IV | 25 |

V | 28 |

VI | 37 |

VII | 39 |

VIII | 41 |

XVI | 105 |

XVII | 107 |

XVIII | 108 |

XIX | 115 |

XX | 123 |

XXI | 124 |

XXII | 130 |

XXIII | 132 |

### Common terms and phrases

aa,p algebraic analytic function arbitrary function Banach spaces boundary value problems bounded domain Cauchy problem characteristic function classical compact Consequently constant converges convex corresponding criterion for nontriviality defined differential equation differential operators Dirichlet problem dual space entire function equality equations of infinite Euclidean space example exists a function exists a number following conditions following inequality formula Fourier transform func function u(x heat equation hm(x imbedding theorems infinite order J. L. LIONS latter inequality Lemma Let us consider linear mathematical monotone moreover Nauk USSR norm obtain operator L(u operators of infinite order 2m ordinary differential equation periodic functions prob problem of infinite proof Remark right side Russian satisfies the conditions sequence MN Sobolev spaces solution solvability space Xm spaces of infinite spaces W”{aa,p sufficient Taking into account Theorem 2.1 theorem is proved theory tion trivial um(x uniformly valid vm(t zero

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Page 4 - As long as algebra and geometry proceeded ics in science ... along separate paths, their advance was slow and their applications limited. Eugene Wigner But when these sciences joined company they drew from each other fresh vitality and Well, if you know of a better 'ole, go to it. thenceforward marched on at a rapid pace towards perfection. Bruce Bairnsfather Joseph Louis Lagrange. What is now proved was once only imagined.

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Page 3 - Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years:...

Page 3 - ... another. The Mathematics and Its Applications programme tries to make available a careful selection of books which fit the philosophy outlined above. With such books, which are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better communication among the practitioners in diversified fields. Because of the wealth of scholarly research being undertaken in the Soviet Union, Eastern Europe, and Japan, it was decided to devote special...