Generalized convolutions and positive definite functions associated with general orthogonal series

Alan Schwartz
1974 Pacific Journal of Mathematics  
Let {φ n } be a sequence of continuous functions orthogonal on an interval with respect to a positive measure da, and let h(n) -(\ \φ n \ z da\ . Then under hypotheses general enough to include as special cases the trigonometric system {e inx } 9 the ultraspherical polynomials, and most cases of the Jacobi polynomials, the sequences <α> satisfying || a ||=Σ~=o I a(n) I h(n)< oo form a Banach algebra with a convolution defined by <α*δ> = where Σ;=o c (n)h(n)φ n = (2£-o a(n)h(ri)φ n )(Σn=o
more » ... )φ n ). Attention is centered upon sequences <α> of unit norm (called distribution sequences), and the associated orthogonal serieŝ Σιd{n)h{n)φ n (called characteristic functions). Theorems on divisibility and stability of these classes are proved, the results being modeled after the corresponding ones about the class of characteristic functions in probability theory.
doi:10.2140/pjm.1974.55.565 fatcat:gctma2vlzfgpxozix4ttqgeity