Generalized Fourier expansions for distributions and ultra distribution

Abstract

Let Dw(Pa)' be a space of distributions or ultradistributions of Beurling type on the p-dimensional parallelepiped Pa:=Ppj=1[-aj,aj]IRp. We investigate the following problems: 1) When can any element of Dw(Pa)' be expanded in absolutely convergent series in a system of generalized exponentials (el(n))nINp with special exponents l(n), nINp. 2) When can a sequence of the coefficients (cn)nINp in an expansionu=SnINp cnel(n) be chosen so that it depends in a continuous and linear way on uIDw(Pa)', where 0<bj£ aj for all 1£j£ p.