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Generalized convolutions and positive definite functions associated with general orthogonal series

1974
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Pacific Journal of Mathematics
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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

doi:10.2140/pjm.1974.55.565
fatcat:gctma2vlzfgpxozix4ttqgeity