Operators preserving orthogonality of polynomials
Marcellán, Francisco ; Szafraniec, Franciszek
Studia Mathematica, Tome 119 (1996), p. 205-218 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

Let S be a degree preserving linear operator of ℝ[X] into itself. The question is if, preserving orthogonality of some orthogonal polynomial sequences, S must necessarily be an operator of composition with some affine function of ℝ. In [2] this problem was considered for S mapping sequences of Laguerre polynomials onto sequences of orthogonal polynomials. Here we improve substantially the theorems of [2] as well as disprove the conjecture proposed there. We also consider the same questions for polynomials orthogonal on the unit circle.

Publié le : 1996-01-01
EUDML-ID : urn:eudml:doc:216332 
@article{bwmeta1.element.bwnjournal-article-smv120i3p205bwm,
     author = {Francisco Marcell\'an and Franciszek Szafraniec},
     title = {Operators preserving orthogonality of polynomials},
     journal = {Studia Mathematica},
     volume = {119},
     year = {1996},
     pages = {205-218},
     zbl = {0861.47018},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv120i3p205bwm}
}

Marcellán, Francisco; Szafraniec, Franciszek. Operators preserving orthogonality of polynomials. Studia Mathematica, Tome 119 (1996) pp. 205-218. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv120i3p205bwm/

[00000] [1] M. Alfaro and Marcellán, F., Recent trends in orthogonal polynomials on the unit circle, in: Orthogonal Polynomials and Their Applications, C. Brezinski, L. Gori and A. Ronveaux, (eds.), IMACS Ann. Comp. Appl. Math. 9, Baltzer, Basel, 1991, 3-14.

[00001] [2] Allaway, W. R., Orthogonality preserving maps and the Laguerre functional, Proc. Amer. Math. Soc. 100 (1987), 82-86. | Zbl 0637.33007

[00002] [3] Askey, R., Orthogonal Polynomials and Special Functions, CBMS-NSF Regional Conf. Ser. in Appl. Math. 21, SIAM, Philadelphia, 1975.

[00003] [4] Geronimus, Ya. L., Polynomials orthogonal on a circle and their applications, Amer. Math. Soc. Transl. 3 (1962), 1-78.

[00004] [5] K. H. Kwon and Littlejohn, L. L., Classification of classical orthogonal polynomials, Utah State University, Research Report 10/93/70. | Zbl 0898.33002

[00005] [6] K. H. Kwon and Littlejohn, The orthogonality of the Laguerre polynomials Ln-k(x) for positive integer k, Ann. Numer. Math. 2 (1995), 289-303. | Zbl 0831.33003

[00006] [7] Szegö, G., Orthogonal Polynomials, Amer. Math. Soc. Providence, R.I., 1975. | Zbl 0305.42011