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Taylor formula for distributions

100%
Ziemian B.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS 0. Introduction................................................................................................................................................................5 1. Preliminary remarks...................................................................................................................................................6 2. Hyperfunctions and their generalizations.................................................................................................................10 3. Flat functions; definitions and properties.................................................................................................................14 4. Taylor formula for quasi $O(x^a)$ functions.............................................................................................................18 5. Homogeneous distributions and their properties......................................................................................................20 6. Mellin transformable distributions.............................................................................................................................22 7. Differential equations in the space of Mellin transformable distributions. Operational calculus for ℳ......................26 8. Taylor formula for distributions.................................................................................................................................29 9. Taylor transformation for functions and distributions................................................................................................32 10. Spectral support of a function and of a distribution................................................................................................33 11. Determination of singularities of solutions of ordinary linear differential operators with smooth coefficients..........36 11. 1. Asymptotic expansion of the push-forward operation $F_{∗}φ$ for F admitting an F-invariant operator...........40 12. Value of a function (distribution) at a point.............................................................................................................41 13. Taylor formula for the product of functions.............................................................................................................44 14. Multiplication of distributions. Taylor formula for the product of distributions..........................................................47 14.1. Spectral topology................................................................................................................................................50 14.2. Heuristic remarks concerning multiplication of distributions.................................................................................51 14.3. Taylor formula for the function 1/f........................................................................................................................51 15. Taylor formula for composite functions...................................................................................................................52 References ..................................................................................................................................................................56
2
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Generalized analytic functions with applications to singular ordinary and partial differential equations

100%
Ziemian B.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS Introduction.............................................................................................................................................5 I. Preliminaries.........................................................................................................................................7    1. A review of classical results in the theory of Laplace integra............................................................7    2. Boundary values of holomorphic functions......................................................................................10      2.1. Distributions as boundary values of holomorphic functions.........................................................10      2.2. Hyperfunctions in one variable....................................................................................................12    3. Mellin analytic functionals, Mellin hyperfunctions and Mellin distributions.........................................14    4. Laplace distributions.........................................................................................................................18      4.1. Convolution of Laplace distributions.............................................................................................21    5. Ecalle distributions.............................................................................................................................23      5.1. Alien derivatives of Ecalle distributions.........................................................................................24    6. Paley-Wiener type theorem for Mellin analytic functionals.................................................................25      6.1. Phragmén-Lindelöf type theorems................................................................................................29    7. The cut-off functions and their Mellin transforms...............................................................................30    8. Modified Cauchy transformation in dimension 1.................................................................................31 II. The theory of generalized analytic functions..........................................................................................33    9. Definition of a generalized analytic function........................................................................................34    10. The Mellin transform of a generalized analytic function.....................................................................35    11. Characterization of GAFs in terms of Mellin transforms.....................................................................37    12. The Borel and Taylor transformations in the class of GAFs..............................................................40    13. Operations on generalized analytic functions...................................................................................40    14. Resurgent functions.........................................................................................................................44      14.1. Alien derivatives of resurgent functions......................................................................................46      14.2. Taylor-Fourier representation of resurgent functions..................................................................47 III. Applications to singular linear differential equations..............................................................................48    15. Special functions as generalized analytic functions...........................................................................48    16. Fuchsian type ODEs with generalized analytic coefficients................................................................52    17. Fuchsian type PDEs with "constant" coefficients................................................................................58    18. GAFs in several variables..................................................................................................................73    19. Fuchsian type PDEs with generalized analytic coefficients................................................................78 Appendices.................................................................................................................................................84 I. The symbol of a distribution in the sense of A. Weinstein. Conormal distributions...................................84 II. Nonlinear singular differential equations.................................................................................................88    1. The case of ordinary differential equations..........................................................................................88    2. The case of partial differential equations.............................................................................................93 References...................................................................................................................................................94 Symbol index.................................................................................................................................................97 Subject index................................................................................................................................................99
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The modified Cauchy transformation with applications to generalized Taylor expansions

95%
Ziemian B.
Studia Mathematica
|
1992
|
tom 102
|
nr 1
1-24
EN
We generalize to the case of several variables the classical theorems on the holomorphic extension of the Cauchy transforms. The Cauchy transformation is considered in the setting of tempered distributions and the Cauchy kernel is modified to a rapidly decreasing function. The results are applied to the study of "continuous" Taylor expansions and to singular partial differential equations.
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Elliptic corner operators in spaces with continuous radial asymptotics II

95%
Ziemian B.
Banach Center Publications
|
1992
|
tom 27
|
nr 2
555-580
EN
Asymptotic expansions at the origin with respect to the radial variable are established for solutions to equations with smooth 2-dimensional singular Fuchsian type operators.
5
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On extendability of invariant distributions

95%
Ziemian B.
Annales Polonici Mathematici
|
2000
|
tom 74
|
nr 1
13-25
EN
In this paper sufficient conditions are given in order that every distribution invariant under a Lie group extend from the set of orbits of maximal dimension to the whole of the space. It is shown that these conditions are satisfied for the n-point action of the pure Lorentz group and for a standard action of the Lorentz group of arbitrary signature.
6
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Topological imbedding of Laplace distributions in Laplace hyperfunctions

66%
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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Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables

60%
Pliś M. E., Ziemian B.
Annales Polonici Mathematici
|
1997
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tom 67
|
nr 1
31-41
EN
We consider a nonlinear Laplace equation Δu = f(x,u) in two variables. Following the methods of B. Braaksma [Br] and J. Ecalle used for some nonlinear ordinary differential equations we construct first a formal power series solution and then we prove the convergence of the series in the same class as the function f in x.
8
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Characterization of Mellin distributions supported by certain noncompact sets

60%
Szmydt Z., Ziemian B.
Studia Mathematica
|
1992
|
tom 102
|
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.
9
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Between the Paley-Wiener theorem and the Bochner tube theorem

60%
Szmydt Z., Ziemian B.
Annales Polonici Mathematici
|
1994-1995
|
tom 60
|
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].
10
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A remark on Nilsson type integrals

60%
Minh N. S., Ziemian B.
Banach Center Publications
|
1996
|
tom 33
|
nr 1
277-285
EN
We investigate ramification properties with respect to parameters of integrals (distributions) of a class of singular functions over an unbounded cycle which may intersect the singularities of the integrand. We generalize the classical result of Nilsson dealing with the case where the cycle is bounded and contained in the set of holomorphy of the integrand. Such problems arise naturally in the study of exponential representation at infinity of solutions to certain PDE's (see [Z]).
11
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An invariance method for constructing fundamental solutions for $P(□_mn)$

48%
Szmydt Z., Ziemian B.
Annales Polonici Mathematici
|
1985
|
tom 46
|
nr 1
333-360
12
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Fundamental solution $E_n$ of the operator $∂^2/∂t^2 - Δ_n$ for n>3

48%
Szmydt Z., Ziemian B.
Annales Polonici Mathematici
|
1979
|
tom 36
|
nr 3
277-286
13
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On distributions invariant with respect to some linear transformations

48%
Ziemian B.
Annales Polonici Mathematici
|
1979
|
tom 36
|
nr 3
261-276
14
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Mean value theorems for linear and semi-linear rotation invariant operators

48%
Ziemian B.
Annales Polonici Mathematici
|
1990
|
tom 51
|
nr 1
341-348
15
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Fundamental solution for operators preserving a quadratic form

42%
Szmydt Z., Ziemian B.
Annales Polonici Mathematici
|
1983
|
tom 42
|
nr 1
369-386
16
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Distributions invariant under compact Lie groups

42%
Ziemian B.
Annales Polonici Mathematici
|
1983
|
tom 43
|
nr 2
175-183
17
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Preface, Contents

31%
Janeczko S., Zajączkowski W. M., Ziemian B.
Banach Center Publications
|
1996
|
tom 33
|
nr 1
1-8
18
Content available remote

Preface, Contents

25%
Jakubczyk B., Janeczko S., Ziemian B.
Banach Center Publications
|
1995
|
tom 34
|
nr 1
1-8
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