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Topological imbedding of Laplace distributions in Laplace hyperfunctions

100%

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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Characterization of Mellin distributions supported by certain noncompact sets

92%

Szmydt Z., Ziemian B.

Studia Mathematica

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1992

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tom 102

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nr 1

25-38

EN

A class of distributions supported by certain noncompact regular sets K are identified with continuous linear functionals on $C_0^∞(K)$. The proof is based on a parameter version of the Seeley extension theorem.

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Between the Paley-Wiener theorem and the Bochner tube theorem

92%

Szmydt Z., Ziemian B.

Annales Polonici Mathematici

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1994-1995

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tom 60

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nr 3

299-304

EN

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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Bogdan Ziemian (1953-1997)

75%

Bojarski B., Łojasiewicz S., Łysik G., Szmydt Z.

Annales Polonici Mathematici

|

2000

|

tom 74

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nr 1

1-11

5

Remarque sur la régularité des intégrales des équations différentielles hyperboliques du second ordre

75%

Szarski J., Szmydt Z., Ważewski T.

Annales Polonici Mathematici

|

1959

|

tom 6

|

nr 3

241-244

6

An invariance method for constructing fundamental solutions for $P(□_mn)$

74%

Szmydt Z., Ziemian B.

Annales Polonici Mathematici

|

1985

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tom 46

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nr 1

333-360

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Fundamental solution $E_n$ of the operator $∂^2/∂t^2 - Δ_n$ for n>3

74%

Szmydt Z., Ziemian B.

Annales Polonici Mathematici

|

1979

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tom 36

|

nr 3

277-286

8

On homogeneous rotation invariant distributions and the Laplace operator

73%

Szmydt Z.

Annales Polonici Mathematici

|

1979

|

tom 36

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nr 3

249-259

9

Fundamental solution for operators preserving a quadratic form

65%

Szmydt Z., Ziemian B.

Annales Polonici Mathematici

|

1983

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tom 42

|

nr 1

369-386

10

Paley-Wiener theorems for the Mellin transformation

64%

Szmydt Z.

Annales Polonici Mathematici

|

1990

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tom 51

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nr 1

313-324

11

Characterization of regular tempered distributions

64%

Szmydt Z.

Annales Polonici Mathematici

|

1983

|

tom 41

|

nr 3

255-258

Ograniczanie wyników

9 Annales Polonici Mathematici

1 Studia Mathematica

11 Szmydt Z.

6 Ziemian B.

1 Bojarski B.

1 Szarski J.

1 Ważewski T.

1 Łojasiewicz S.

1 Łysik G.

1 2000

1 1998

1 1995

1 1992

1 1990

1 1985

2 1983

2 1979

1 1959

Topological imbedding of Laplace distributions in Laplace hyperfunctions

Characterization of Mellin distributions supported by certain noncompact sets

Between the Paley-Wiener theorem and the Bochner tube theorem

Bogdan Ziemian (1953-1997)

Remarque sur la régularité des intégrales des équations différentielles hyperboliques du second ordre

An invariance method for constructing fundamental solutions for $P(□_mn)$

Fundamental solution $E_n$ of the operator $∂^2/∂t^2 - Δ_n$ for n>3

On homogeneous rotation invariant distributions and the Laplace operator

Fundamental solution for operators preserving a quadratic form

Paley-Wiener theorems for the Mellin transformation

Characterization of regular tempered distributions