Generalized Fourier expansions for distributions and ultradistributions (Q1974104)

The expansions of distributions and ultradistributions in generalized exponential series are well known. Let \({\mathcal D}_w(\Pi_a)_\beta'\) be the space of ultradistributions of Beurling type on \(p\)-dimensional parallelepiped \(\Pi_a:= \prod^p_{j=1} [-a_j, a_j]\subset \mathbb{R}^p\) with its strong topology. Let \(\lambda_{(n)}:= (\lambda^{(j)}_{n_j})^p_{j= 1}\) be a sequence of exponents with \((\lambda^{(j)}_m)_{m\in\mathbb{N}}\) being simple zeros of an entire function \(L_j\). In this paper the author obtains a necessary and sufficient condition for the following problems. 1. When can any \(w\)-ultra differentiable function on \(\Pi_a\) or any \(u\in{\mathcal D}_w(\Pi_a)_\beta'\) can be expanded in a generalized Fourier series in the system \((e_{\lambda_{(n)}})_{n\in\mathbb{N}^p}\) absolutely convergent in \({\mathcal D}_w(\Pi_a)_\beta'\). 2. When a sequence of coefficients \(c= (c_n)_{n\in \mathbb{N}^p}\) in an expansion \(u= \sum_{n\in \mathbb{N}^p} c_ne_{\lambda_{(n)}}\) be chosen so that it depends in a continuous linear way on \(u\). In other words, when does the map \(c\mapsto \sum_{n\in \mathbb{N}^p}c_n e_{\lambda_{(n)}}\) admit a continuous linear right inverse.