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1996 | 120 | 3 | 205-218
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Operators preserving orthogonality of polynomials

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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.
Twórcy
  • Escuela Politécnica Superior, Universidad Carlos III de Madrid, E-28911 Leganés, c/Butarque, 15, Spain, pacomarc@ing.uc3m.es
  • Instytut Matematyki, Uniwersytet Jagielloński, ul. Reymonta 4, PL-30059 Kraków, Poland, fhszafra@im.uj.edu.pl
Bibliografia
  • [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.
  • [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.
  • [6] K. H. Kwon and Littlejohn, The orthogonality of the Laguerre polynomials $L_n^-k(x)$ for positive integer k, Ann. Numer. Math. 2 (1995), 289-303.
  • [7] Szegö, G., Orthogonal Polynomials, Amer. Math. Soc. Providence, R.I., 1975.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv120i3p205bwm
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