CONTENTS0. Introduction................................................................................................................................................................51. Preliminary remarks...................................................................................................................................................62. Hyperfunctions and their generalizations.................................................................................................................103....
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...
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.
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.
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.
CONTENTS Foreword..............................................................................................................................5 Introduction..........................................................................................................................6 1. Preliminaries....................................................................................................................7 1.1. Terminology and notation.............................................................................................7...
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]).
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].
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.
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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