# Operators preserving orthogonality of polynomials

Francisco Marcellán; Franciszek Szafraniec

Studia Mathematica (1996)

- Volume: 120, Issue: 3, page 205-218
- ISSN: 0039-3223

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topMarcellán, Francisco, and Szafraniec, Franciszek. "Operators preserving orthogonality of polynomials." Studia Mathematica 120.3 (1996): 205-218. <http://eudml.org/doc/216332>.

@article{Marcellán1996,

abstract = {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.},

author = {Marcellán, Francisco, Szafraniec, Franciszek},

journal = {Studia Mathematica},

keywords = {Laguerre polynomials; polynomials orthogonal on the unit circle; linear operators preserving orthogonality; operators preserving orthogonality of polynomials; operator of composition},

language = {eng},

number = {3},

pages = {205-218},

title = {Operators preserving orthogonality of polynomials},

url = {http://eudml.org/doc/216332},

volume = {120},

year = {1996},

}

TY - JOUR

AU - Marcellán, Francisco

AU - Szafraniec, Franciszek

TI - Operators preserving orthogonality of polynomials

JO - Studia Mathematica

PY - 1996

VL - 120

IS - 3

SP - 205

EP - 218

AB - 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.

LA - eng

KW - Laguerre polynomials; polynomials orthogonal on the unit circle; linear operators preserving orthogonality; operators preserving orthogonality of polynomials; operator of composition

UR - http://eudml.org/doc/216332

ER -

## References

top- [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.
- [2] Allaway, W. R., Orthogonality preserving maps and the Laguerre functional, Proc. Amer. Math. Soc. 100 (1987), 82-86. Zbl0637.33007
- [3] Askey, R., Orthogonal Polynomials and Special Functions, CBMS-NSF Regional Conf. Ser. in Appl. Math. 21, SIAM, Philadelphia, 1975.
- [4] Geronimus, Ya. L., Polynomials orthogonal on a circle and their applications, Amer. Math. Soc. Transl. 3 (1962), 1-78.
- [5] K. H. Kwon and Littlejohn, L. L., Classification of classical orthogonal polynomials, Utah State University, Research Report 10/93/70. Zbl0898.33002
- [6] K. H. Kwon and Littlejohn, The orthogonality of the Laguerre polynomials ${L}_{n}^{-}k\left(x\right)$ for positive integer k, Ann. Numer. Math. 2 (1995), 289-303. Zbl0831.33003
- [7] Szegö, G., Orthogonal Polynomials, Amer. Math. Soc. Providence, R.I., 1975. Zbl0305.42011

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