Asymptotic formulae for the zeros of orthogonal polynomials

V. M. Badkov

Sbornik. Mathematics, Tome 203 (2012) no. 9, pp. 1231-1243

Résumé

Let $p_n(t)$ be an algebraic polynomial that is orthonormal with weight $p(t)$ on the interval $[-1, 1]$. When $p(t)$ is a perturbation (in certain limits) of the Chebyshev weight of the first kind, the zeros of the polynomial $p_n(\cos\tau)$ and the differences between pairs of (not necessarily consecutive) zeros are shown to satisfy asymptotic formulae as $n\to\infty$, which hold uniformly with respect to the indices of the zeros. Similar results are also obtained for perturbations of the Chebyshev weight of the second kind. First, some preliminary results on the asymptotic behaviour of the difference between two zeros of an orthogonal trigonometric polynomial, which are needed, are established. Bibliography: 15 titles.

Détail

Keywords: zeros, asymptotic formulae.
Mots-clés : orthogonal polynomials

Bibliographie

[1] Eu. Seneta, Regularly varying functions, Lecture Notes in Math., 508, Springer-Verlag, Berlin–Heidelberg–New York, 1976 | DOI | MR | MR | Zbl | Zbl

[2] G. Szegő, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., 23, Amer. Math. Soc., Providence, RI, 1959 | MR | Zbl | Zbl

[3] S. N. Bernshtein, “O mnogochlenakh, ortogonalnykh na konechnom otrezke”, Sobranie sochinenii. Konstruktivnaya teoriya funktsii, v. 2, Izd-vo AN SSSR, M., 1954, 7–106 | MR | Zbl

[4] V. M. Badkov, “Zeros of orthogonal polynomials”, Function theory, Proc. Inst. Math. Mech., suppl. 2, 2005, S30–S48 | MR | Zbl

[5] V. M. Badkov, “Approximations of functions in the uniform metric by Fourier sums of orthogonal polynomials”, Proc. Steklov Inst. Math., 145 (1981), 19–65 | MR | Zbl | Zbl

[6] P. G. Nevai, “An asymptotic formula for the derivatives of orthogonal polynomials”, SIAM J. Math. Anal., 10:3 (1979), 472–477 | DOI | MR | Zbl

[7] N. K. Bary, A treatise on trigonometric series, Macmillan, New York, 1964 | MR | MR | Zbl

[8] Ya. L. Geronimus, Polynomials orthogonal on a circle and interval, Pergamon Press, Oxford–London–New York–Paris, 1961 | MR | MR | Zbl | Zbl

[9] V. M. Badkov, “Asymptotic and extremal properties of orthogonal polynomials in the presence of singularities in the weight”, Proc. Steklov Inst. Math., 198 (1994), 37–82 | MR | Zbl | Zbl

[10] V. M. Badkov, “Ravnomernye asimptoticheskie predstavleniya ortogonalnykh mnogochlenov”, Priblizhenie funktsii polinomami i splainami, UNTs AN SSSR, Sverdlovsk, 1985, 41–53 | MR | Zbl

[11] V. M. Badkov, “Uniform asymptotic representations of orthogonal polynomials and their derivatives”, Proc. Steklov Inst. Math., 180 (1989), 39–41 | Zbl

[12] V. M. Badkov, “Ob osobennostyakh vesa, otnositelno kotorogo ortogonalny mnogochleny vtorogo roda”, Izv. TulGU. Estestv. nauki, 1, Izd-vo TulGU, Tula, 2008, 6–18

[13] B. L. Golinskii, “Utochnenie asimptoticheskikh formul G. Segë i S. N. Bernshteina”, Izv. vuzov. Matem., 1968, no. 11, 70–82 | MR | Zbl

[14] P. K. Suetin, “Fundamental properties of polynomials orthogonal on a contour”, Russian Math. Surveys, 21:2 (1966), 35–83 | DOI | MR | Zbl

[15] V. M. Badkov, “The asymptotic behavior of orthogonal polynomials”, Math. USSR-Sb., 37:1 (1980), 39–51 | DOI | MR | Zbl | Zbl