Mellin analysis of singular and non-linear PDE's
Taylor formula for distributions
CONTENTS0. Introduction................................................................................................................................................................51. Preliminary remarks...................................................................................................................................................62. Hyperfunctions and their generalizations.................................................................................................................103....
Generalized analytic functions with applications to singular ordinary and partial differential equations
CONTENTS Introduction.............................................................................................................................................5 I. Preliminaries.........................................................................................................................................7 1. A review of classical results in the theory of Laplace integra............................................................7 2. Boundary values of holomorphic functions......................................................................................10...
Distributions invariant under compact Lie groups
On distributions invariant with respect to some linear transformations
On extendability of invariant distributions
In this paper sufficient conditions are given in order that every distribution invariant under a Lie group extend from the set of orbits of maximal dimension to the whole of the space. It is shown that these conditions are satisfied for the n-point action of the pure Lorentz group and for a standard action of the Lorentz group of arbitrary signature.
Elliptic corner operators in spaces with continuous radial asymptotics II
Asymptotic expansions at the origin with respect to the radial variable are established for solutions to equations with smooth 2-dimensional singular Fuchsian type operators.
The modified Cauchy transformation with applications to generalized Taylor expansions
We generalize to the case of several variables the classical theorems on the holomorphic extension of the Cauchy transforms. The Cauchy transformation is considered in the setting of tempered distributions and the Cauchy kernel is modified to a rapidly decreasing function. The results are applied to the study of "continuous" Taylor expansions and to singular partial differential equations.
Mean value theorems for linear and semi-linear rotation invariant operators
Topological imbedding of Laplace distributions in Laplace hyperfunctions
CONTENTS Foreword..............................................................................................................................5 Introduction..........................................................................................................................6 1. Preliminaries....................................................................................................................7 1.1. Terminology and notation.............................................................................................7...
An invariance method for constructing fundamental solutions for
Fundamental solution of the operator for n>3
Fundamental solution for operators preserving a quadratic form
A remark on Nilsson type integrals
We investigate ramification properties with respect to parameters of integrals (distributions) of a class of singular functions over an unbounded cycle which may intersect the singularities of the integrand. We generalize the classical result of Nilsson dealing with the case where the cycle is bounded and contained in the set of holomorphy of the integrand. Such problems arise naturally in the study of exponential representation at infinity of solutions to certain PDE's (see [Z]).
Between the Paley-Wiener theorem and the Bochner tube theorem
We present the classical Paley-Wiener-Schwartz theorem [1] on the Laplace transform of a compactly supported distribution in a new framework which arises naturally in the study of the Mellin transformation. In particular, sufficient conditions for a function to be the Mellin (Laplace) transform of a compactly supported distribution are given in the form resembling the Bochner tube theorem [2].
Characterization of Mellin distributions supported by certain noncompact sets
A class of distributions supported by certain noncompact regular sets K are identified with continuous linear functionals on . The proof is based on a parameter version of the Seeley extension theorem.
Preface, Contents
Borel resummation of formal solutions to nonlinear Laplace equations in 2 variables
We consider a nonlinear Laplace equation Δu = f(x,u) in two variables. Following the methods of B. Braaksma [Br] and J. Ecalle used for some nonlinear ordinary differential equations we construct first a formal power series solution and then we prove the convergence of the series in the same class as the function f in x.
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