Some ramifications of a theorem of Boas and Pollard concerning the completion of a set of functions in

About fifty years ago, R. P. Boas and Harry Pollard proved that an orthonormal system that is completable by the adjunction of a finite number of functions also can be completed by multiplying the elements of the given system by a fixed, bounded, nonnegative measurable function. In subsequent years, several variations and extensions of this theorem have been given by a number of other investigators, and this program is continued here. A mildly surprising corollary of one of the results is that the trigonometric and Walsh systems can be multiplicatively transformed into quasibases for .