Singularly continuous measures in Nevai’s class $M$
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- by D. S. Lubinsky PDF
- Proc. Amer. Math. Soc. 111 (1991), 413-420 Request permission
Abstract:
Let $d\nu$ be a nonnegative Borel measure on $[ - \pi ,\pi ]$, with $0 < \smallint _{ - \pi }^\pi d\nu < \infty$ and with support of Lebesgue measure zero. We show that there exist $\{ {\eta _j}\} _{j = 1}^\infty \subset (0,\infty )$ and $\{ {t_j}\} _{j = 1}^\infty \subset ( - \pi ,\pi )$ such that if \[ d\mu (\theta ): = \sum \limits _{j = 1}^\infty {{\eta _j}d\nu (\theta + {t_j}),\quad \theta \in [ - \pi ,\pi ],} \](with the usual periodic extension $d\nu (\theta \pm 2\pi ) = d\nu (\theta )$), then the leading coefficients $\{ {\kappa _n}(d\mu )\} _{n = 0}^\infty$ of the orthonormal polynomials for $d\mu$ satisfy \[ \lim \limits _{n \to \infty } {\kappa _n}(d\mu )/{\kappa _{n + 1}}(d\mu ) = 1.\] As a consequence, we obtain pure singularly continuous measures $d\alpha$ on $[ - 1,1]$ lying in Nevai’s class $M$.References
- François Delyon, Barry Simon, and Bernard Souillard, From power pure point to continuous spectrum in disordered systems, Ann. Inst. H. Poincaré Phys. Théor. 42 (1985), no. 3, 283–309 (English, with French summary). MR 797277
- M. F. Barnsley, J. S. Geronimo, and A. N. Harrington, Almost periodic Jacobi matrices associated with Julia sets for polynomials, Comm. Math. Phys. 99 (1985), no. 3, 303–317. MR 795106, DOI 10.1007/BF01240350
- D. Bessis, J. S. Geronimo, and P. Moussa, Function weighted measures and orthogonal polynomials on Julia sets, Constr. Approx. 4 (1988), no. 2, 157–173. MR 932652, DOI 10.1007/BF02075456
- D. S. Lubinsky, A survey of general orthogonal polynomials for weights on finite and infinite intervals, Acta Appl. Math. 10 (1987), no. 3, 237–296. MR 920673, DOI 10.1007/BF00049120
- D. S. Lubinsky, Jump distributions on $[-1,1]$ whose orthogonal polynomials have leading coefficients with given asymptotic behavior, Proc. Amer. Math. Soc. 104 (1988), no. 2, 516–524. MR 962822, DOI 10.1090/S0002-9939-1988-0962822-2
- Walter Van Assche and Alphonse P. Magnus, Sieved orthogonal polynomials and discrete measures with jumps dense in an interval, Proc. Amer. Math. Soc. 106 (1989), no. 1, 163–173. MR 953001, DOI 10.1090/S0002-9939-1989-0953001-4
- Attila Máté, Paul Nevai, and Vilmos Totik, Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle, Constr. Approx. 1 (1985), no. 1, 63–69. MR 766095, DOI 10.1007/BF01890022
- H. N. Mhaskar and E. B. Saff, On the distribution of zeros of polynomials orthogonal on the unit circle, J. Approx. Theory 63 (1990), no. 1, 30–38. MR 1074079, DOI 10.1016/0021-9045(90)90111-3
- Paul G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185. MR 519926, DOI 10.1090/memo/0213
- Paul Nevai, Géza Freud, orthogonal polynomials and Christoffel functions. A case study, J. Approx. Theory 48 (1986), no. 1, 3–167. MR 862231, DOI 10.1016/0021-9045(86)90016-X
- Paul Nevai and Vilmos Totik, Orthogonal polynomials and their zeros, Acta Sci. Math. (Szeged) 53 (1989), no. 1-2, 99–104. MR 1018677 E. A. Rahmanov, On the asymptotics of the ratio of orthogonal polynomials, Math. USSR Sbornik 32 (1977), 199-213.
- E. B. Saff, Orthogonal polynomials from a complex perspective, Orthogonal polynomials (Columbus, OH, 1989) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 294, Kluwer Acad. Publ., Dordrecht, 1990, pp. 363–393. MR 1100302, DOI 10.1007/978-94-009-0501-6_{1}7
- Herbert Stahl and Vilmos Totik, $n$th root asymptotic behavior of orthonormal polynomials, Orthogonal polynomials (Columbus, OH, 1989) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 294, Kluwer Acad. Publ., Dordrecht, 1990, pp. 395–417. MR 1100303 G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1939, 4th ed., 1975.
- J. L. Ullman and M. F. Wyneken, Weak limits of zeros of orthogonal polynomials, Constr. Approx. 2 (1986), no. 4, 339–347. MR 892160, DOI 10.1007/BF01893436
- Walter Van Assche, Asymptotics for orthogonal polynomials, Lecture Notes in Mathematics, vol. 1265, Springer-Verlag, Berlin, 1987. MR 903848, DOI 10.1007/BFb0081880
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 111 (1991), 413-420
- MSC: Primary 42C05; Secondary 39A10
- DOI: https://doi.org/10.1090/S0002-9939-1991-1039259-3
- MathSciNet review: 1039259