\(p\)-frames and shift invariant subspaces of \(L^p\) (Q5931798)

This paper is devoted to the study of necessary and sufficient conditions for an indexed family \(\{\Phi_i\}\) to construct a \(p\)-frame for \(V_p(\Phi)\), and to generate a closed shift invariant space of \(L^p\). The note contains very interesting parts 1 and 2 (introduction and preliminaries) and moreover a part with lemmas (pp. 9-14). The main results are conditions given in Theorem 1 which are equivalent to each other. Proofs of these implications which are full and long are presented in part 5.