The Asymptotic Zero Distribution of Orthogonal Polynomials with Varying Recurrence Coefficients

Journal of Approximation Theory

Volumn 99, Issue 1, 1999, Pages 167-197

  Kuijlaars, A B J ^a   Van Assche, W ^b  

^a CITY UNIVERSITY OF HONG KONG   (Hong Kong)

^b UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001114597     PISSN: 00219045     EISSN: None     Source Type: Journal    
DOI: 10.1006/jath.1999.3316     Document Type: Article

References (38)

1. Chen L.-C., Ismail M. E. H. On asymptotics of Jacobi polynomials. SIAM J. Math. Anal. 22:1991;1442-1449.
2. Chihara T. S. An Introduction to Orthogonal Polynomials. 1978;Gordon & Breach, New York.
3. Deift P., McLaughlin K. T.-R. A continuum limit of the Toda lattice. Mem. Amer. Math. Soc. 131:1998.
4. Dette H., Studden W. Some new asymptotic properties for the zeros of Jacobi, Laguerre and Hermite polynomials. Constr. Approx. 11:1995;227-238.
5. Dragnev P. D., Saff E. B. Constrained energy problems with applications to orthogonal polynomials of a discrete variable. J. Anal. Math. 72:1997;223-259.
6. P. D. Dragnev, and, E. B. Saff, A problem in potential theory and zero asymptotics of Krawtchouk polynomials, J. Approx. Theory, (to appear).
7. Erdos P., Turán P. On interpolation. III. Interpolatory theory of polynomials. Ann. of Math. 41:1940;510-553.
8. Faldey J., Gawronski W. On the limit distributions of the zeros of Jonquière polynomials and generalized classical orthogonal polynomials. J. Approx. Theory. 81:1995;231-249.
9. Gawronski W. On the asymptotic distribution of the zeros of Hermite, Laguerre, and Jonquière polynomials. J. Approx. Theory. 50:1987;214-231.
10. n. Analysis. 13:1993;29-67.
11. n. Nevai P., Pinkus A. Progress in Approximation Theory. 1991;379-404 Academic Press, New York.
12. Goh W. M. Y., Wimp J. On the asymptotics of the Tricomi-Carlitz polynomials and their zero distributions. SIAM J. Math. Anal. 25:1994;420-428.
13. Goh W. M. Y., Wimp J. The zero distribution of the Tricomi-Carlitz polynomials. Comp. Math. Appl. 33:1997;119-127.
14. Gonchar A. A., Rakhmanov E. A. Equilibrium measure and the distribution of zeros of extremal polynomials. Mat. Sb. 25:1984;117-127.
15. Ismail M. E. H., Li X. Bounds on the extreme zeros of orthogonal polynomials. Proc. Amer. Math. Soc. 115:1992;131-140.
16. Koekoek R., Swarttouw R. F. Report. 1998.
17. Kuijlaars A. B. J., Van Assche W. Extremal polynomials on discrete sets. Proc. London Math. Soc. 79:1999;191-221.
18. Lubinsky D. S., Mhaskar H. N., Saff E. B. A proof of Freud's conjecture for exponential weights. Constr. Approx. 4:1988;65-83.
19. Maté A., Nevai P. Remarks on E. A. Rakhmanov's paper "On the Asymptotics of the Ratio of Orthogonal Polynomials" J. Approx. Theory. 36:1982;64-72.
20. Mhaskar H. N., Saff E. B. Extremal problems for polynomials with exponential weights. Trans. Amer. Math. Soc. 285:1984;203-234.
21. Mhaskar H. N., Saff E. B. Where does the sup-norm of a weighted polynomial live? Constr. Approx. 1:1985;71-91.
22. n)(x). Trans. Amer. Math. Soc. 249:1979;159-162.
23. Nevai P. Orthogonal polynomials. Mem. Amer. Math. Soc. 213:1979.
24. Nevai P., Van Assche W. Remarks on the (C, -1)-summability of the distribution of zeros of orthogonal polynomials. Proc. Amer. Math. Soc. 122:1994;759-767.
25. Pastur L. A. Spectral and probabilistic aspects of matrix models. Boutet de Monvel A., Marchenko V. Algebraic and Geometric Methods in Mathematical Physics. 1996;207-242 Kluwer Academic, Dordrecht.
26. Rakhmanov E. A. On the asymptotics of the ratio of orthogonal polynomials. Mat. Sb. 103:1977;237-252.
27. Rakhmanov E. A. On the asymptotics of the ratio of orthogonal polynomials, II. Mat. Sb. 118:1982;104-117.
28. Rakhmanov E. A. On asymptotics properties of polynomials orthogonal on the real axis. Mat. Sb. 119:1982;163-203.
29. Rakhmanov E. A. Equilibrium measure and the distribution of zeros of the extremal polynomials of a discrete variable. Mat. Sb. 187:1996;109-124.
30. Saff E. B., Totik V. Logarithmic Potentials with External Fields. 1997;Springer-Verlag, Berlin.
31. Saff E. B., Ullman J. L., Varga R. S. Incomplete polynomials: An electrostatics approach. Cheney E. W. Approximation Theory, III. 1980;769-782.
32. Stahl H., Totik V. General Orthogonal Polynomials. 1992;Cambridge Univ. Press, Cambridge.
33. Szego G. Orthogonal Polynomials. 1975;Amer. Math. Soc. Providence.
34. Van Assche W. Asymptotics for Orthogonal Polynomials. Lecture Notes in Mathematics. 1265:1987;Springer-Verlag, Berlin.
35. Van Assche W. Asymptotic properties of orthogonal polynomials from their recurrence formula, II. J. Approx. Theory. 52:1988;322-338.
36. Van Assche W. Norm behavior and zero distribution for orthogonal polynomials with non-symmetric exponential weights. Constr. Approx. 5:1989;329-345.
37. Van Assche W. Asymptotics for orthogonal polynomials and three-term recurrences. Nevai P. Orthogonal Polynomials. NATO ASI Series C. 294:1990;435-462 Kluwer Academic, Dordrecht.
38. Van Assche W., Koornwinder T. H. Asymptotic behaviour for Wall polynomials and the addition formula for little q-Legendre polynomials. SIAM J. Math. Anal. 22:1991;302-311.