Orthonormal basis functions for continuous-time systems and \(L_p\) convergence
The authors provide an analysis of the approximation properties of certain general classes of rational orthonormal basis functions. These bases generalize the Laguerre and Kautz cases. It is proved that the model sets are everywhere dense in the Hardy space \(H_1(\Pi)\) under the same conditions as those derived by the authors for denseness in \(H_p(\Pi)\), \(1< p< \infty\). As an extension, it is shown how to construct orthonormal model sets with prescribed asymptotic order that are everywhere dense in \(H_p(\Pi)\), \(1\leq p< \infty\). It is also shown that the Fourier series formed with the above basis converge in all spaces \(H_p(\Pi)\) and \(H_p(D)\), where \(D\) is the open unit disk and \(1< p< \infty\).
scientific article; zbMATH DE number 1348038