Topological imbedding of Laplace distributions in Laplace hyperfunctions

Zofia Szmydt, and Bogdan Ziemian. Topological imbedding of Laplace distributions in Laplace hyperfunctions. 1998. <http://eudml.org/doc/271241>.

@book{ZofiaSzmydt1998,
abstract = {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 $(D_K)^\{\prime \}$..............................................................................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 $(D^k(K))^\{\prime \}$.......................................................................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 $D^\{\prime \}_K ↪ B_K$, 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........................................................................................................................811991 Mathematics Subject Classification: 46F12, 46F15, 46F20.},
author = {Zofia Szmydt, Bogdan Ziemian},
keywords = {Laplace distributions; Laplace hyperfunctions; Mellin distributions; Mellin hyperfunctions; boundary values; holomorphic functions; Laplace analytic functionals; inductive limit},
language = {eng},
title = {Topological imbedding of Laplace distributions in Laplace hyperfunctions},
url = {http://eudml.org/doc/271241},
year = {1998},
}

TY - BOOK
AU - Zofia Szmydt
AU - Bogdan Ziemian
TI - Topological imbedding of Laplace distributions in Laplace hyperfunctions
PY - 1998
AB - 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 $(D_K)^{\prime }$..............................................................................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 $(D^k(K))^{\prime }$.......................................................................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 $D^{\prime }_K ↪ B_K$, 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........................................................................................................................811991 Mathematics Subject Classification: 46F12, 46F15, 46F20.
LA - eng
KW - Laplace distributions; Laplace hyperfunctions; Mellin distributions; Mellin hyperfunctions; boundary values; holomorphic functions; Laplace analytic functionals; inductive limit
UR - http://eudml.org/doc/271241
ER -