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Poincaré theorem and nonlinear PDE's

100%
Pliś M.
Annales Polonici Mathematici
|
1998
|
tom 69
|
nr 2
99-105
EN
A family of formal solutions of some type of nonlinear partial differential equations is found. Terms of such solutions are Laplace transforms of some Laplace distributions. The series of these distributions are locally finite.
2
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Generalized analytic functions of Bogdan Ziemian

100%
Łysik G.
Annales Polonici Mathematici
|
2000
|
tom 74
|
nr 1
35-41
EN
Definitions, properties, examples and applications of generalized analytic functions introduced by B. Ziemian are presented.
3
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Convolution equations in the space of Laplace distributions

100%
Pliś M.
Annales Polonici Mathematici
|
1998
|
tom 69
|
nr 3
271-281
EN
A formal solution of a nonlinear equation P(D)u = g(u) in 2 variables is constructed using the Laplace transformation and a convolution equation. We assume some conditions on the characteristic set Char P.
4
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A Phragmén-Lindelöf type quasi-analyticity principle

100%
Łysik G.
Studia Mathematica
|
1997
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tom 123
|
nr 3
217-234
EN
Quasi-analyticity theorems of Phragmén-Lindelöf type for holomorphic functions of exponential type on a half plane are stated and proved. Spaces of Laplace distributions (ultradistributions) on ℝ are studied and their boundary value representation is given. A generalization of the Painlevé theorem is proved.
5
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Topological imbedding of Laplace distributions in Laplace hyperfunctions

90%
Szmydt Z., Ziemian B.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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
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