Let {ϕn} be a sequence of
continuous functions orthogonal on an interval with respect to a positive measure dα,
and let h(n) = (∫ab|ϕn|2dα)−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 ⟨a⟩ satisfying
∥a∥ =∑n=0∞|a(n)|h(n) < ∞ form a Banach algebra with a convolution defined by
⟨a∗b⟩ = ⟨c⟩ where ∑n=0∞c(n)h(n)ϕn= (∑n=0∞a(n)h(n)ϕn)(∑n=0∞b(n)h(n)ϕn).
Attention is centered upon sequences ⟨a⟩ of 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.
