Vol. 35, No. 3, 1970

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The derived set of the spectrum of a distribution function

Theodore Seio Chihara

Vol. 35 (1970), No. 3, 571–574
Abstract

Let {an}n=0 and {bn}n=0 be real sequences with bn > 0, bn 0(n →∞). Let {Pn(αj)}n=0 be the sequence of orthonormal polynomials satisfying the recurrence

xPn (x) = bn−1Pn−1(x)+ anPn(x)+ bnPn+1(x),(n ≧ 0),

P− 1(x) = 0,P0(x) = 1.

Then there is a substantially unique distribution function ψ with respect to which the Pn(x) are orthogonal. This paper verifies a conjecture of D. P. Maki that the set of all limit points of the sequence {an} is the derived set of the spectrum of ψ.

Mathematical Subject Classification
Primary: 42.15
Milestones
Received: 9 March 1970
Published: 1 December 1970
Authors
Theodore Seio Chihara