Vol. 55, No. 2, 1974

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Generalized convolutions and positive definite functions associated with general orthogonal series

Alan Schwartz

Vol. 55 (1974), No. 2, 565–582

Let {ϕn} be a sequence of continuous functions orthogonal on an interval with respect to a positive measure , and let h(n) = ( ab|ϕn|2 )1. Then under hypotheses general enough to include as special cases the trigonometric system {einx}, the ultraspherical polynomials, and most cases of the Jacobi polynomials, the sequences asatisfying a= n=0|a(n)|h(n) < form a Banach algebra with a convolution defined by ab= cwhere n=0c(n)h(n)ϕn = ( n=0a(n)h(n)ϕn)( n=0b(n)h(n)ϕn). Attention is centered upon sequences aof unit norm (called distribution sequences), and the associated orthogonal series a(n)h(n)ϕn (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.

Mathematical Subject Classification 2000
Primary: 42A88
Secondary: 60E05, 42A60
Received: 12 March 1973
Revised: 13 March 1974
Published: 1 December 1974
Alan Schwartz