Geodesic
Sets of Polynomials Orthogonal Simultaneously on Four Ellipses
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 1281-1294

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Goodman, Ruth. Sets of Polynomials Orthogonal Simultaneously on Four Ellipses. Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 1281-1294. doi: 10.4153/CJM-1968-126-9
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[1] 1. Merriman, G. M., Concerning sets of polynomials orthogonal simultaneously on several curves Bull. Amer. Math. Soc. 44(1938), 57–69. Google Scholar

[2] 2. Szegö, G., A problem concerning orthogonal polynomials, Trans. Amer. Math. Soc. 37 (1935), 196–206. Google Scholar

[3] 3. Walsh, J. L., Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. Colloq. Publ., Vol. 20, p. 134 (Amer. Math. Soc, Providence, Rhode Island, 1935).10.1090/coll/020 Google Scholar | DOI

[4] 4. Walsh, J. L. and Merriman, G. M., Note on the simultaneous orthogonality of harmonic polynomials on several curves, Duke Math. J. 3 (1937), 279–288. Google Scholar

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