Analytic Pseudo-Differential Operators and their Applications [Hardcover]

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 l£)