Part I develops the theory of pseudodifferential operators with real analytic symbols, the local representatives of which are linear differential operators of infinite order acting in the spaces of basic and generalized functions based on the duality of the spaces of real analytic functions and functionals. Applications to a variety of problems of PDEs and numerical analysis are given. Part II is devoted to the theory of Sobolev-Orlicz spaces of infinite order and the solvability of nonlinear partial differential equations with arbitrary nonlinearities.

Contents:

• Preliminaries
• Pseudo-Differential Operators with Real Analytic Symbols
• Applications to Pseudo-Differential Equations
• Approximation Methods
• A Mollification Method for Ill-Posed Problems
• Nontriviality of Sobolev-Orlicz Spaces of Infinite Order
• Some Properties of Sobolev-Orlicz Spaces of Infinite Order
• Elliptic Equations of Infinite Order with Arbitrary Nonlinearities