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A Criterion for the Nonuniqueness of the Measure of Orthogonality

✍ Scribed by Eyangelos K. Ifantis

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
260 KB
Volume
89
Category
Article
ISSN
0021-9045

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✦ Synopsis

Let the polynomials P n (x), n 1, be defned by P 0 (x)=0, P 1 (x)=1, a n P n+1 (x)+ a n&1 P n&1 (x)+b n P n (x)=xP n (x), n 1. If a n >0 and b n are real then there exists at least one measure of orthogonality for the polynomials P n (x), n=1, 2, ... . The problem of finding conditions on the sequences a n and b n under which this measure is unique or nonunique still remains open for large classes of sequences a n and b n .

Here a new criterion for the nonuniqueness of the measure of orthogonality is proved. This was achieived by proving that the infinite-dimensional Jacobi matrix associated with the sequences a n and b n is not self-adjoint.

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