Representation of Distributions and Ultradistributions by Holomorphic Functions

Abstract

Publisher Summary This chapter focuses on the representation of distributions and ultradistributions by holomorphic functions.The general idea consists in taking smaller spaces of test functions than D (Ω) i n such a way that the essential statements of the theory remain true (in a generalized form), while the dual spaces are enlarged. Beurling defines spaces of test functions by growth properties of their Fourier transforms, while Roumieu uses classes of ultradifferentiable functions taken from classical analysis. Ultradifferentiable functions in Gevrey classes appear in quite a natural way in the theory of partial differential operators with constant coefficients. Regarding the representation of ultradistributions by boundary values of holomorphic functions, Komatsu derives deep necessary and sufficient results of local nature.