Analytic Pseudo-Differential Operators and Their Applications

ISBN-13: 9780792312963 / Angielski / Twarda / 1991 / 252 str.

Iu A. Dubinskii; Julii A. Dubinski

Opis

One service mathematics has rendered the 'Et moi, ..., si j'avait su comment en revenir, je n'y serais point aIle: ' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ... '; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics ... '. All arguably true. And all statements obtainable this way form part of the raison d'elre of this series.

Spis treści

I. PD-Operators with Complex Arguments.- 1. PD-Operators with Constant Analytic Symbols.- 1.1. Spaces of entire functions of exponential type.- 1. The space Exp R(Czn) (5). 2. Estimates of derivatives (7). 3. The testfunction space Exp?(Cnz). Topology and convergence (8). 4. Density of linear combinations of exp ?z, ? ? ? (12)..- 1.2. PD-operators with analytic symbols.- 1. Local algebra of differential operators of infinite order (15). 2. Algebra of PD-operators with arbitrary analytic symbols (20). 3. The correctness of the definition of a PD-operator (23)..- 1.3. The operator method.- 1. PD-equations in the whole space Cn (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.- 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 exponential functionals (47)..- 2. Fourier Transformation of Arbitrary Analytic Functions. Complex Fourier Method.- 2.1. Fourier transformation.- 1. Main definition. The inversion formula (52). 2. The Fourier image of exponential functions. The Borel kernel (54). 3. Complex unitarity (58)..- 2.2. Complex Fourier method.- 1. Table of duality. Examples (59). 2. Fourier method for PD-equations (62)..- 3. PD-Equations whose Symbols are Formal Series.- 3.1. Differential operators of infinite order with constant coefficients.- 1. The space Eq,r Czn of entire functions of order q order with variable coefficients.- 1. Definition of a d.o.i.o. with variable coefficients (76). 2. The Cauchy problem in the spaces Eq,rCzn (78). 3. The Cauchy problem in the spaces Eq,r+? Czn.- II. The Cauchy Problem in the Complex Domain.- 4. Cauchy-Kovalevskaya Theory in Spaces of Analytic Functions with Pole-type Singularities.- 4.1. The Cauchy problem in the spaces Dm,r (Case of cylindrical evolution).- 1. The test-function space Dm,r (90). 2. Criterion for the well-posedness of the Cauchy problem in the spaces Dm,r (91). 3. The structure of systems with ord Aij ? mi — mj (96). 4. The Cauchy problem in the dual spaces D’m,r (98)..- 4.2. The Cauchy problem in the spaces Dm,r_?.- 1. The test-function space Dmr_?t.- 4.3. Formulation of the basic results for arbitrary systems in normal form.- 5. Exponential theory of the Cauchy problem.- 5.1. The Cauchy problem in the scale of spaces of initial data.- 1. Banach spaces of entire functions of finite order (120). 2. Criterion for the well-posedness of the Cauchy problem in the scale Expm,r,q( Czn) (123). 3. The structure of systems satisfying the conditions deg {IEqu1 ix-1}..- 5.2. The Cauchy problem in the scale of “linearly increasing” spaces of initial data.- 1. The scale Expm,r+?.- 2. Criterion for the well-posedness of the Cauchy problem in the scale Expm,r+?.- 6. PD-Operators with Variable Analytic Symbols.- 6.1. Basic definitions.- 1. Definition of the PD-operator A (z,D) (157). 2. Definitions of the requisite spaces (159)..- 6.2. The Cauchy problem in the scale Expm,r(Czz).- 6.3. The Cauchy problem in the scale Expm,r+?.- Conclusion: The connection between the Cauchy exponential theory and the classical Cauchy-Kovalevskaya theory.- III. PD-Operators with Real Arguments.- 7. Spaces of Test Functions and Distributions.- 7.1. The test-function space H?(SR).- 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).- 1. Definition of H-221E; (SR). Main property (182). 2. Examples of functionals in H-?(SR) (185)..- 7.3. Sobolev spaces of infinite order W?{a?},p (Rn).- 1. Criterion for non-triviality of the spaces W?{a?,p}(Rn) (188). 2. The distribution space W-?{a?,p?}(Rn) (193)..- 8. Analytic PD-operators with Real Arguments. Applications.- 8.1. Algebra of PD-operators with analytic symbols.- 1. The space H?(G) (196). 2. The action of PD-operators (198). 3. Examples (201). 4. The dual theory (203). 5. A possible generalization (205)..- 8.2. PD-equations in the whole Euclidean space.- 8.3. The Cauchy problem.- 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.- 1. The Cauchy problem for a homogeneous equation (214). 2. Special case. Laplace equation (216). 3. The Cauchy problem for the heat equation (218). 4. One equation with a shift (218). 5. Quasipolynomial 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 continuation of a pair of functions defined on ?1 (225)..- 8.5. Quantum relativistic particle with zero spin.- 1. Derivation of the Schrödinger equation (226). 2. Fundamental solution of the Cauchy problem (230). 3. Lorentz invariance (233). 4. Description 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.- Author Index.- Index of Basic Formulae.