Matrix Mean Series in Terms of Boundary Orthogonal Systems and Functions in the Classes H and E

We give conditions for orthonormal systems on the boundary of a plane Jordan domain which are necessary and sufficient for an arbitrary series in terms of this orthonormal system to be the Fourier series of some function in H^∞(G) resp. E^P(G)(1<p<∞). Our results contain a classical criterion of Fejér for the boundedness of a holomorphic function in the unit disk.