Differential Operations Of Infinite Order With Real Arguments And Their Applications

Part 1 Preliminaries: Convolution of Distributions; Fourier Transforms of Distributions; Entire Functions of Exponential Type that are Bounded on Rn; The Dirichlet Kernels; Markov Type Theorems; The Second Interpolation Method of Bernstein; Orlicz Classes and Orlicz Spaces. Part 2 Pseudodifferential Operators with Real Analytic Symbols: The Space of Test Functions in a Neighbourhood of Zero; The Differential Operators of Infinite Order. Part 3 Applications to Pseudodifferential Equations: Boundary-Value Problems; Multi-Dimensional Integral Equations of the First Kind with Entire Kernel. Part 4 Approximation Methods: Approximating the Symbols by Algebraic Polynomials; Approximating the Symbols by Trigonometric Polynomials; Approximate the Data by Sinc Functions. Part 5 Mollification Methods: The Heat Equation Backwards in Time; The Cauchy Problem for the Laplace Equation; The Noncharacteristic Cauchy Problem for Parabolic Equations. Part 6 Sobolev-Orlicz Spaces of Infinite Order: Limits of the Monotone Sequence of Banach Spaces; The Nontriviality of Sobolov-Orlicz Spaces of Infinite Order; Embeddings of Sobolev-Orlicz Spaces of Infinite Order. Part 7 Nonlinear Differential Equations of Infinite Order: Elliptic Boundary Value Problems of Infinite Order. (Part contents).