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Between the Paley-Wiener theorem and the Bochner tube theorem

100%

Szmydt Z., Ziemian B.

Annales Polonici Mathematici

1994-1995

tom 60

nr 3

299-304

We present the classical Paley-Wiener-Schwartz theorem [1] on the Laplace transform of a compactly supported distribution in a new framework which arises naturally in the study of the Mellin transformation. In particular, sufficient conditions for a function to be the Mellin (Laplace) transform of a compactly supported distribution are given in the form resembling the Bochner tube theorem [2].

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Topological imbedding of Laplace distributions in Laplace hyperfunctions

72%

Szmydt Z., Ziemian B.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS Foreword..............................................................................................................................5 Introduction..........................................................................................................................6 1. Preliminaries....................................................................................................................7 1.1. Terminology and notation.............................................................................................7 1.2. Selected topics on complex topological vector spaces and their duals........................7 2. A review of basic facts in the theory of distributions.......................................................10 2.1. The spaces $D_K$ and $(D_K)'$..............................................................................10 2.2. The spaces D(Ω) and D'(Ω).......................................................................................11 2.3. The spaces D(A) and D'(A)........................................................................................12 2.4. The spaces $D^k(K)$ and $(D^k(K))'$.......................................................................14 3. Selected topics in the theory of holomorphic functions of one variable..........................15 3.1. Basic notions and theorems.......................................................................................15 3.2. The spaces A(K) and A'(K)........................................................................................16 3.3. Boundary values of holomorphic functions of one variable........................................19 4. Hyperfunctions in one variable.......................................................................................23 4.1. Definitions and basic properties of hyperfunctions....................................................23 4.2. Imbedding of analytic functions in hyperfunctions: A(Ω) ↪ B(Ω)................................26 4.3. Elementary operations on hyperfunctions..................................................................27 4.4. The Köthe theorem....................................................................................................28 4.5. Imbedding $D'_K ↪ B_K$, K compact in ℝ................................................................31 4.6. The distributional version of the Köthe theorem........................................................34 4.7. Imbedding D'(Ω)↪ B(Ω), Ω open in ℝ........................................................................35 4.8. Hyperfunctional boundary values of holomorphic functions.......................................39 4.9. Hyperfunctions supported by a single point...............................................................39 4.10. Substitution in a hyperfunction and in an analytic functional (distribution)...............40 5. Laplace hyperfunctions and Laplace analytic functionals in one variable......................42 6. Mellin hyperfunctions and Mellin distributions in one variable........................................50 6.1. Mellin hyperfunctions and Mellin analytic functionals.................................................50 6.2. Mellin distributions.....................................................................................................60 7. Laplace distributions L'(ω)(ℝ͞͞₊)......................................................................................64 7.1. Definitions and basic properties of Laplace distributions...........................................64 7.2. Imbedding of Laplace distributions in Laplace hyperfunctions...................................67 7.3. Imbedding of Mellin distributions in Mellin hyperfunctions..........................................79 References........................................................................................................................81

Ograniczanie wyników

1 Annales Polonici Mathematici

2 Szmydt Z.

2 Ziemian B.

1 1998

1 1995

Wyniki wyszukiwania

Between the Paley-Wiener theorem and the Bochner tube theorem

Topological imbedding of Laplace distributions in Laplace hyperfunctions