Sets of Polynomials Orthogonal Simultaneously on Four Ellipses

It has been shown by Walsh (3) and Szegö (2) that if a set of polynomials is orthogonal on both of two distinct curves, then one curve is a level curve of the other. Szegö (2) has determined all sets of polynomials which are orthogonal simultaneously on an entire family of level curves. There are fi...

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Bibliographic Details
Published in:Canadian journal of mathematics 1968, Vol.20, p.1281-1294
Main Author: Goodman, Ruth
Format: Article
Language:English
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Summary:It has been shown by Walsh (3) and Szegö (2) that if a set of polynomials is orthogonal on both of two distinct curves, then one curve is a level curve of the other. Szegö (2) has determined all sets of polynomials which are orthogonal simultaneously on an entire family of level curves. There are five essentially different sets, two of which are orthogonal on concentric circles, and three of which are orthogonal on confocal ellipses. Merriman (1) has shown that the orthogonality of a set of polynomials on both of two concentric circles is sufficient to guarantee their orthogonality on the entire family of circles.
ISSN:0008-414X
1496-4279
DOI:10.4153/CJM-1968-126-9