Analytic pseudo-differential operators and their applications:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer
1991
|
| Series: | Mathematics and its applications / Soviet series
68 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus d. Russ. übers. |
| Physical Description: | XII, 252 S. |
| ISBN: | 0792312961 |
Staff View
MARC
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| 100 | 1 | |a Dubinskij, Julij A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Analytic pseudo-differential operators and their applications |c by Julii A. Dubinskii |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 1991 | |
| 300 | |a XII, 252 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications / Soviet series |v 68 | |
| 500 | |a Aus d. Russ. übers. | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
SERIES EDITOR S PREFACE v
PREFACE xi
PART I. PD Operators with Complex Arguments 1
Introduction 1
CHAPTER 1. PD Operators with Constant Analytic Symbols ... 5
1.1. Spaces of entire functions of exponential type 5
1. The space Expfi(C?) (5). 2. Estimates of derivatives (7). 3. The test
function space Expn(C ). Topology and convergence (8). 4. Density
of linear combinations of exp Xz, Aeft (12).
1.2. PD operators with analytic symbols 15
1. Local algebra of differential operators of infinite order (15). 2. Al¬
gebra of PD operators with arbitrary analytic symbols (20). 3. The
correctness of the definition of a PD operator (23).
1.3. The operator method 28
1. PD equations in the whole space C (28). 2. The Cauchy problem
in the space of exponential functions (30). 3. Cauchy Kovalevskaya
theorem (special case) (35). 4. A two point boundary value problem
(40).
1.4. The dual theory 41
1. Exponential functionals. Examples (41). 2. The general form of
exponential functionals (43). 3. The algebra of PD operators in the
space of exponential functionals (45). 4. Cauchy problem in exponen¬
tial functionals (47).
CHAPTER 2. Fourier Transformation of Arbitrary Analytic
Functions. Complex Fourier Method 52
2.1. Fourier transformation 52
1. Main definition. The inversion formula (52). 2. The Fourier image
of exponential functions. The Borel kernel (54). 3. Complex unitarity
(58).
viii CONTENTS
2.2. Complex Fourier method 59
1. Table of duality. Examples (59). 2. Fourier method for PD equations
(62).
CHAPTER 3. PD Equations whose Symbols are Formal Series . . 66
3.1. Differential operators of infinite order with constant coefficients ... 66
1. The space £,,r(CJ) of entire functions of order q 1 (66). 2. The
basic estimate (67). 3. The non formal action of d.o.i.o. s (70). 4. The
algebra of d.o.i.o. s (72). 5. The Cauchy problem (73).
3.2. Differential operators of infinite order with variable coefficients ... 76
1. Definition of a d.o.i.o. with variable coefficients (76). 2. The Cauchy
problem in the spaces EtiT{CV:) (78). 3. The Cauchy problem in the
spaces £,ir+a|«|(C?) (80).
PART II. The Cauchy Problem in the Complex Domain 85
Introduction 85
CHAPTER 4. Cauchy Kovalevskaya Theory in Spaces of Analytic
Functions with Pole type Singularities 90
4.1. The Cauchy problem in the spaces Dmr (Case of cylindrical
evolution) 90
1. The test function space Dmr (90). 2. Criterion for the well posedness
of the Cauchy problem in the spaces Dm,r (91). 3. The structure of
systems with ordj4tJ m, — rrij (96). 4. The Cauchy problem in the
dual spaces D m r (98).
4.2. The Cauchy problem in the spaces Dmr_a i . (Case of conical
evolution) 101
1. The test function space Dmr_a t (101). 2. Criterion for the well
posedness of the Cauchy problem in the spaces Dmr_a^ . Necessity of
Kovalevskaya conditions and Leray Volevich conditions (102). 3. The
structure of systems with oxdAij m* — rrij¦, + 1 (106). 4. The Cauchy
problem in the dual scale D m+1 r+ 7i i (HO).
4.3. Formulation of the basic results for arbitrary systems in normal form 117
CHAPTER 5. Exponential theory of the Cauchy problem .... 120
5.1. The Cauchy problem in the scale of spaces of initial data 120
1. Banach spaces of entire functions of finite order (120). 2. Cri¬
terion for the well posedness of the Cauchy problem in the scale
CONTENTS ix
Expm r ?(C ) (123). 3. The structure of systems satisfying the con¬
ditions dega? m, rtij a (q 1) (129).
5.2. The Cauchy problem in the scale of linearly increasing spaces
of initial data 133
1. The scale Expmr+CT|(|g(C ) of entire functions of finite order (133).
2. Criterion for the well posedness of the Cauchy problem in the scale
Expm r+ ,|t|,«,(C?) (134). 3. The structure of systems with dega?
m, — rrij — a (q — 1) + q (141). 4. The Cauchy problem in the dual
space Exp^^n^C?) (151).
CHAPTER 6. PD Operators with Variable Analytic Symbols . . 157
6.1. Basic definitions 157
1. Definition of the PD operator A(z,V) (157). 2. Definitions of the
requisite spaces (159).
6.2. The Cauchy problem in the scale Expm r(C£) 160
6.3. The Cauchy problem in the scale Expmr+ T|,|(C^) 166
Conclusion: The connection between the Cauchy exponential theory and
the classical Cauchy Kovalevskaya theory 171
PART III. PD Operators with Real Arguments 174
Introduction 174
CHAPTER 7. Spaces of Test Functions and Distributions .... 176
7.1. The test function space H°°(Sr) 176
1. Definition and examples of test functions (176). 2. Description of
H°°(SR) in the x variables (178). 3. Convergence in H°°{SR) (179).
4. Invariance of H°°(Sr) under differential operators of infinite order
(180). 5. An example (181).
7.2. The generalized function space H °°(SR) 182
1. Definition of H~CO(SR). Main property (182). 2. Examples of func
tionals in H °°(SR) (185).
7.3. Sobolev spaces of infinite order W°°{aa,p}(Rn) 187
1. Criterion for non triviality of the spaces W°°{aa,p}(Rn) (188).
2. The distribution space W °°{aa,p }(Rn) (193).
CHAPTER 8. Analytic PD operators with Real Arguments.
Applications 196
8.1. Algebra of PD operators with analytic symbols 196
x CONTENTS
1. The space H°°(G) (196). 2. The action of PD operators (198).
3. Examples (201). 4. The dual theory (203). 5. A possible gener¬
alization (205).
8.2. PD equations in the whole Euclidean space 206
8.3. The Cauchy problem 208
1. The Cauchy problem in the space H°°(G) (208). 2. The Cauchy
problem in the dual space H~°°(G) (211). 3. On the existence of the
fundamental solution of the Cauchy problem (213).
8.4. Examples 214
1. The Cauchy problem for a homogeneous equation (214). 2. Spe¬
cial case. Laplace equation (216). 3. The Cauchy problem for the
heat equation (218). 4. One equation with a shift (218). 5. Quasi
polynomial solutions (219). 6. One boundary value problem in a strip
(220). 7. The boundary value problem in the cylinder (221). 8. The
Dirichlet problem in a disc. Poisson integral (222). 9. The Dirichlet
problem in the half plane. Cauchy integral (224). 10. Analytic contin¬
uation of a pair of functions defined on R1 (225).
8.5. Quantum relativistic particle with zero spin 226
1. Derivation of the Schrodinger equation (226). 2. Fundamental solu¬
tion of the Cauchy problem (230). 3. Lorentz invariance (233). 4. De¬
scription of Lorentz invariant solutions (236). 5. Recurrence formulae
for the Lorentz invariant solutions (238). 6. Non relativistic limit and
factorization of Klein Gordon Fock operator (239).
References 242
Author index 248
Index of basic formulae 250
Subject index 251
|
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| author_facet | Dubinskij, Julij A. |
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| bvnumber | BV004807929 |
| classification_rvk | SK 620 |
| ctrlnum | (OCoLC)246569998 (DE-599)BVBBV004807929 |
| discipline | Mathematik |
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| id | DE-604.BV004807929 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T16:17:57Z |
| institution | BVB |
| isbn | 0792312961 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002957735 |
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| owner_facet | DE-12 DE-739 DE-29T DE-703 DE-83 DE-11 DE-188 |
| physical | XII, 252 S. |
| publishDate | 1991 |
| publishDateSearch | 1991 |
| publishDateSort | 1991 |
| publisher | Kluwer |
| record_format | marc |
| series2 | Mathematics and its applications / Soviet series |
| spelling | Dubinskij, Julij A. Verfasser aut Analytic pseudo-differential operators and their applications by Julii A. Dubinskii Dordrecht [u.a.] Kluwer 1991 XII, 252 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications / Soviet series 68 Aus d. Russ. übers. Pseudodifferentialoperator (DE-588)4047640-6 gnd rswk-swf Pseudodifferentialoperator (DE-588)4047640-6 s DE-604 Soviet series Mathematics and its applications 68 (DE-604)BV004708148 68 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002957735&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dubinskij, Julij A. Analytic pseudo-differential operators and their applications Pseudodifferentialoperator (DE-588)4047640-6 gnd |
| subject_GND | (DE-588)4047640-6 |
| title | Analytic pseudo-differential operators and their applications |
| title_auth | Analytic pseudo-differential operators and their applications |
| title_exact_search | Analytic pseudo-differential operators and their applications |
| title_full | Analytic pseudo-differential operators and their applications by Julii A. Dubinskii |
| title_fullStr | Analytic pseudo-differential operators and their applications by Julii A. Dubinskii |
| title_full_unstemmed | Analytic pseudo-differential operators and their applications by Julii A. Dubinskii |
| title_short | Analytic pseudo-differential operators and their applications |
| title_sort | analytic pseudo differential operators and their applications |
| topic | Pseudodifferentialoperator (DE-588)4047640-6 gnd |
| topic_facet | Pseudodifferentialoperator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002957735&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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