Multidimensional Tauberian Theorem for Holomorphic Functions of Bounded Argument

Abstract

Generalized functions with a Laplace transform having a bounded argument in a tube domain over the positive orthant are considered. Necessary and sufficient conditions for the existence of quasi-asymptotics of such functions are found. A regularly varying function with respect to which such quasi-asymptotics exist is explicitly given. It turns out that the modulus of a holomorphic function in a tube domain over the positive orthant in the purely imaginary subspace on rays entering the origin behaves as a regularly varying function. The obtained results are used to find quasi-asymptotics of solutions to the generalized Cauchy problem for convolution equations with kernels being passive operators. Multidimensional passive operators and systems are often encountered in mathematical physics. Examples are operators that are hyperbolic with respect to a cone, transport equations, equations for complex electric circuits without energy pumping, the Maxwell and Dirac equations, etc.