Bases of exponentials in spaces \(E^ p(D^ n)\) on a polyhedron and representation of functions of that space in form of a sum of periodic ones (Q916836)

Let \(D_ 1,...,D_ n\subset {\mathbb{C}}\) be open convex polygons containing the origin, and let \(D=D_ 1\times...\times D_ n\). Let \(s(D_ j)\) be the family of sequences \(\{C_{\nu}\}_{\nu \in {\mathbb{N}}}\subset D_ j\) of closed rectifiable curves such that \(C_{\nu}\to \partial D_ j\) as \(\nu\to \infty\). Suppose \(1\leq p<\infty\). One defines the space \(E^ p(D)\) as follows. A holomorphic function f: \(D\to {\mathbb{C}}\) is said to belong to \(E^ p(D)\) if for any \(\{C_{\nu_ j}^{(j)}\}_{\nu_ j\in {\mathbb{N}}}\in s(D_ j)\), \(j=1,...,n\), one has \[ \sup_{\nu}\int_{C_{\nu}}| f(z)|^ p| dz_ 1|...| dz_ n| <\infty, \] where \(C_{\nu}=C_{\nu_ 1}^{(1)}\times...\times C_{\nu_ n}^{(n)}\) and the supremum is taken over all multiindices \(\nu =(\nu_ 1,...,\nu_ n)\) in \({\mathbb{N}}^ n\). The author constructs and investigates bases for \(E^ p(D)\), consisting of functions that are finite sums of functions of the form \(z\mapsto a \exp (<z,b>)\).