Linear Differential Operators with Constant Coefficients

Title Preface Table of Contents Introduction § 1. Exponential representation for an ordinary equation with one unknown function § 2. Exponential representation of the solutions of partial differential equations § 3. The exponential representation of solutions of arbitrary systems Part One: Analytic Methods I. Homological Tools § 1. Families of topological modules § 2. The fundamental homology theorem § 3. Operations on modules II. Division with Remainder in the Space of Power Series § 1. The space of power series § 2. The base sequence of matrices § 3. Stabilization of the base sequence § 4. p-decompositions III. Cohomologies of Analytic Functions of Bounded Growth § 1. The space of holomorphic functions § 2. The operator Dz in spaces of type I § 3. M-cohomologies § 4. The theorem on the triviality of M-cohomologies § 5. Cohomologies connected with P-matrices IV. The Fundamental Theorem § 1. Some properties of finite P-modules § 2. Local p-operators § 3. The fundamental inequality for the operator D § 4. Noetherian operators § 5. The fundamental theorem Part Two: Differential Equations with Constant Coefficients V. Linear Spaces and Distributions § 1. Limiting processes in families of linear spaces § 2. Functional spaces § 3. Fourier transform VI. Homogeneous Systems of Equations § 4. The exponential representation of solutions of homogeneous systems of equations § 5. Hypoelliptic operators § 6. Uniqueness of solutions of the Cauchy problem VII. Inhomogeneous Systems § 7. Solubility of inhomogeneous systems. M-convexity § 8. M-convexity in convex regions § 9. The connection between M-convexity and the properties of a sheaf of solutions of a homogeneous system § 10. The algebraic condition for M-convexity § 11. Geometrical conditions of M-convexity § 12. Operators of the form p(Dxi) in domains of holomorphy VIII. Overdetermined Systems § 13. Concerning the modules Ext^i(M, P) § 14. The extension of solutions of homogeneous systems § 15. The influence of boundary values on the behavior of the solutions within a region Notes Bibliography Subject Index Index of Basic Notation