Linear Differential Operators with Constant Coefficients

✍ Scribed by Victor P. Palamodov

• Publisher: Springer
• Year: 1970
• Tongue: English
• Leaves: 458
• Series: Die Grundlagen der mathematischen Wissenschaften in Einzeldarstellungen #168
• Category: Library

✦ Table of Contents

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