Topological imbedding of Laplace distributions in Laplace hyperfunctions

Szmydt Zofia; Ziemian Bogdan

  • Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1998

Abstract

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CONTENTSForeword..............................................................................................................................5Introduction..........................................................................................................................61. Preliminaries....................................................................................................................7 1.1. Terminology and notation.............................................................................................7 1.2. Selected topics on complex topological vector spaces and their duals........................72. 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 ) ) ' .......................................................................143. 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........................................194. 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)...............405. Laplace hyperfunctions and Laplace analytic functionals in one variable......................426. Mellin hyperfunctions and Mellin distributions in one variable........................................50 6.1. Mellin hyperfunctions and Mellin analytic functionals.................................................50 6.2. Mellin distributions.....................................................................................................607. 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..........................................79References........................................................................................................................811991 Mathematics Subject Classification: 46F12, 46F15, 46F20.

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

@book{SzmydtZofia1998,
abstract = {CONTENTSForeword..............................................................................................................................5Introduction..........................................................................................................................61. Preliminaries....................................................................................................................7 1.1. Terminology and notation.............................................................................................7 1.2. Selected topics on complex topological vector spaces and their duals........................72. 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 $(D^k(K))^\{\prime \}$.......................................................................143. 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........................................194. 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)...............405. Laplace hyperfunctions and Laplace analytic functionals in one variable......................426. Mellin hyperfunctions and Mellin distributions in one variable........................................50 6.1. Mellin hyperfunctions and Mellin analytic functionals.................................................50 6.2. Mellin distributions.....................................................................................................607. 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..........................................79References........................................................................................................................811991 Mathematics Subject Classification: 46F12, 46F15, 46F20.},
author = {Szmydt Zofia, Ziemian Bogdan},
keywords = {Laplace distributions; Laplace hyperfunctions; Mellin distributions; Mellin hyperfunctions; boundary values; holomorphic functions; Laplace analytic functionals; inductive limit},
language = {eng},
location = {Warszawa},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Topological imbedding of Laplace distributions in Laplace hyperfunctions},
url = {http://eudml.org/doc/271241},
year = {1998},
}

TY - BOOK
AU - Szmydt Zofia
AU - Ziemian Bogdan
TI - Topological imbedding of Laplace distributions in Laplace hyperfunctions
PY - 1998
CY - Warszawa
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - CONTENTSForeword..............................................................................................................................5Introduction..........................................................................................................................61. Preliminaries....................................................................................................................7 1.1. Terminology and notation.............................................................................................7 1.2. Selected topics on complex topological vector spaces and their duals........................72. 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 $(D^k(K))^{\prime }$.......................................................................143. 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........................................194. 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)...............405. Laplace hyperfunctions and Laplace analytic functionals in one variable......................426. Mellin hyperfunctions and Mellin distributions in one variable........................................50 6.1. Mellin hyperfunctions and Mellin analytic functionals.................................................50 6.2. Mellin distributions.....................................................................................................607. 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..........................................79References........................................................................................................................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 -

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