Sieved orthogonal polynomials and discrete measures with jumps dense in an interval
Authors:
Walter Van Assche and Alphonse P. Magnus
Journal:
Proc. Amer. Math. Soc. 106 (1989), 163173
MSC:
Primary 42C05; Secondary 33A65
MathSciNet review:
953001
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Abstract: We investigate particular classes of sieved Jacobi polynomials for which the weight function vanishes at the zeros of a Chebyshev polynomial of the first kind. These polynomials are then used to give a proof, using only orthogonal polynomials on , that the discrete orthogonal polynomials introduced by Lubinsky have converging recurrence coefficients. We construct similar discrete measures with jumps dense in and use sieved ultraspherical polynomials to show that their recurrence coefficients converge.
 [AAA]
Waleed
AlSalam, W.
R. Allaway, and Richard
Askey, Sieved ultraspherical
polynomials, Trans. Amer. Math. Soc.
284 (1984), no. 1,
39–55. MR
742411 (85j:33005), http://dx.doi.org/10.1090/S00029947198407424116
 [C]
T.
S. Chihara, An introduction to orthogonal polynomials, Gordon
and Breach Science Publishers, New YorkLondonParis, 1978. Mathematics and
its Applications, Vol. 13. MR 0481884
(58 #1979)
 [CI]
Jairo
Charris and Mourad
E. H. Ismail, On sieved orthogonal polynomials. II. Random walk
polynomials, Canad. J. Math. 38 (1986), no. 2,
397–415. MR
833576 (87j:33014a), http://dx.doi.org/10.4153/CJM1986020x
 [DSS]
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 (87d:35098)
 [GVA]
J.
S. Geronimo and W.
Van Assche, Orthogonal polynomials on several
intervals via a polynomial mapping, Trans.
Amer. Math. Soc. 308 (1988), no. 2, 559–581. MR 951620
(89f:42021), http://dx.doi.org/10.1090/S00029947198809516206
 [I1]
Mourad
E. H. Ismail, On sieved orthogonal polynomials. I. Symmetric
Pollaczek analogues, SIAM J. Math. Anal. 16 (1985),
no. 5, 1093–1113. MR 800799
(87a:33019), http://dx.doi.org/10.1137/0516081
 [I2]
Mourad
E. H. Ismail, On sieved orthogonal polynomials. III.
Orthogonality on several intervals, Trans.
Amer. Math. Soc. 294 (1986), no. 1, 89–111. MR 819937
(87j:33014b), http://dx.doi.org/10.1090/S00029947198608199371
 [I3]
Mourad
E. H. Ismail, On sieved orthogonal polynomials. IV. Generating
functions, J. Approx. Theory 46 (1986), no. 3,
284–296. MR
840396 (87j:33014c), http://dx.doi.org/10.1016/00219045(86)900663
 [L]
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 (90c:42031), http://dx.doi.org/10.1090/S00029939198809628222
 [MNT]
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
(85j:42045), http://dx.doi.org/10.1007/BF01890022
 [N1]
Paul
G. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc.
18 (1979), no. 213, v+185. MR 519926
(80k:42025), http://dx.doi.org/10.1090/memo/0213
 [N2]
Paul
Nevai, Mean convergence of Lagrange
interpolation. III, Trans. Amer. Math. Soc.
282 (1984), no. 2,
669–698. MR
732113 (85c:41009), http://dx.doi.org/10.1090/S00029947198407321134
 [R1]
E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, Mat. Sb. 103 (1977), 237252; Math. USSRSb. 32 (1977), 199213.
 [R2]
, On the asymptotics of the ratio of orthogonal polynomials, II, Mat. Sb. 118 (1982), 104117; Math. USSRSb. 46 (1983), 105117.
 [S]
G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 23, Providence, R.I., 4th ed., 1975.
 [AAA]
 W. AlSalam, W. Allaway, and R. Askey, Sieved ultraspherical polynomials, Trans. Amer. Math. Soc. 284 (1984), 3955. MR 742411 (85j:33005)
 [C]
 T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, New York, 1978. MR 0481884 (58:1979)
 [CI]
 J. Charris and M. E. H. Ismail, On sieved orthogonal polynomials, II: random walk polynomials, Canad. J. Math. 38 (1986), 397415. MR 833576 (87j:33014a)
 [DSS]
 F. Delyon, B. Simon, and B. Souillard, From power pure point to continuous spectrum in disordered systems, Ann. Inst. H. Poincaré, Phys. Théor. 42 (1985), 283309. MR 797277 (87d:35098)
 [GVA]
 J. S. Geronimo and W. Van Assche, Orthogonal polynomials on several intervals via a polynomial mapping, Trans. Amer. Math. Soc. 308 (1988), 559581. MR 951620 (89f:42021)
 [I1]
 M. E. H. Ismail, On sieved orthogonal polynomials, I: symmetric Pollaczek analogues, SIAM J. Math. Anal. 16 (1985), 10931113. MR 800799 (87a:33019)
 [I2]
 , On sieved orthogonal polynomials, III: orthogonality on several intervals, Trans. Amer. Math. Soc.294 (1986), 89111. MR 819937 (87j:33014b)
 [I3]
 , On sieved orthogonal polynomials, IV: generating functions, J. Approx. Theory 46 (1986), 284296. MR 840396 (87j:33014c)
 [L]
 D. S. Lubinsky, Jump distributions on whose orthogonal polynomials have leading coefficients with given asymptotic behaviour, Proc. Amer. Math. Soc. 104 (1988), 516524. MR 962822 (90c:42031)
 [MNT]
 A. Máté, P. Nevai, and V. Totik, Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle, Constr. Approx. 1 (1985), 6369. MR 766095 (85j:42045)
 [N1]
 P. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 213, Providence, R.I., 1979. MR 519926 (80k:42025)
 [N2]
 , Mean convergence of Lagrange interpolation, III, Trans. Amer. Math. Soc. 282 (1984), 669698. MR 732113 (85c:41009)
 [R1]
 E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, Mat. Sb. 103 (1977), 237252; Math. USSRSb. 32 (1977), 199213.
 [R2]
 , On the asymptotics of the ratio of orthogonal polynomials, II, Mat. Sb. 118 (1982), 104117; Math. USSRSb. 46 (1983), 105117.
 [S]
 G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ. 23, Providence, R.I., 4th ed., 1975.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198909530014
PII:
S 00029939(1989)09530014
Keywords:
Sieved orthogonal polynomials,
Chebyshev polynomials,
Nevai's class
Article copyright:
© Copyright 1989
American Mathematical Society
