Treatise on Analysis

Treatise on Analysis, Volume 10–VII provides information pertinent to the fundamental aspects of linear functional equations. This book discusses the problems dealing with functional equations of scalar or of vectors.

Comprised of one chapter, this volume begins with a description and study of the primary concepts and tools that have prompted the progress in the study of linear partial differential equations. This text then explains the importance of the integral operators. The reader is also introduced to integral operators that operate not only on vector function, but also on sections of vector bundles. This book discusses as well the applications of the differential operators to spectral theory.

This book is a valuable resource for mathematicians.

Notation

Chapter XXIII Linear Functional Equations

Part One : Pseudodifferential Operators

1. Integral Operators

2. Integral Operators of Proper Type

3. Integral Operators on Vector Bundles

4. Density Bundle and Kernel Sections

5. Bounded Sections

6. Volterra Operators

7. Carleman Operators

8. Generalized Eigenfunctions

9. Kernel Distributions

10. Regular Kernel Distributions

11. Smoothing Operators and Composition of Operators

12. Wave Front of a Distribution

13. Convolution Equations

14. Elementary Solutions

15. Existence and Uniqueness Problems for Systems of Linear Partial Differential Equations

16. Operator Symbols

17. Oscillating Integrals

18. Lax-Maslov Operators

19. Pseudodifferential Operators

20. Symbol of a Pseudodifferential Operator of Proper Type

21. Matrix Pseudodifferential Operators

22. Parametrix of Elliptical Operators on an Open Subset of Rn

23. Pseudodifferential Operators in Hs0(X) Spaces

24. Classical Dirichlet Problem and Coarse Dirichlet Problem

25. The Green Operator

26. Pseudodifferential Operators on a Manifold

27. Adjoint of a Pseudodifferential Operator on a Manifold. Composition of Two Pseudodifferential Operators on a Manifold

28. Extension of Pseudodifferential Operators to Distribution Sections

29. Principal Symbols

30. Parametrix of Elliptic Operators on Manifolds

31. Spectral Theory of Hermitian Elliptic Operators : I. Self-Adjoint Extensions and Boundary Conditions

32. Spectral Theory of Hermitian Elliptic Operators : II. Generalized Eigenfunctions

33. Essentially Self-Adjoint Pseudodifferential Operators : I. Hermitian Convolution Operators on Rn

34. Essentially Self-Adjoint Pseudodifferential Operators : II. Atomic Spectra

35. Essentially Self-Adjoint Pseudodifferential Operators : III. Hermitian Elliptic Operators on a Compact Manifold

36. Invariant Differential Operators

37. Differential Properties of Spherical Functions

38. Example : Spherical Harmonics

References

Index