Inverse images of polynomial mappings and polynomials orthogonal on them

Journal of Computational and Applied Mathematics

Volumn 153, Issue 1-2, 2003, Pages 371-385

  Peherstorfer, Franz ^a  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

Author keywords

Functionals; Inverse images; Orthogonal polynomials; Polynomial mappings; Positive definite; Positive measure

Indexed keywords

MAPPING; NUMERICAL ANALYSIS; POLYNOMIALS;

POLYNOMIAL MAPPINGS;

COMPUTATIONAL METHODS;

MATHEMATICAL ANALYSIS;

EID: 0037383435     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(02)00628-3     Document Type: Article

References (28)

1. W. Al-Salam, W. Allaway, R. Askey, Sieved ultraspherical polynomials, Trans. Amer. Math. Soc. 284 (1984) 39-55.
2. D. Barrios, G. Lopéz, E. Torrano, Polynomials generated by asymptotically periodic complex recurrence coefficients, Math. USSR Sb. 186 (1995) 629-659.
3. B. Beckermann, Complex Jacobi matrices, J. Comput. Appl. Math. 127 (2001) 17-65.
4. D. Bessis, Orthogonal polynomials, Padé approximations and Julia sets, in: P. Nevai (Ed.), Orthogonal Polynomials: Theory and Practice, NATO ASI Series, Series C: Mathematical and Physical Sciences, Vol. 294, Kluwer Academic, Dordrecht, Boston, London, 1990, pp. 55-97.
5. D. Bessis, P. Moussa, Orthogonality properties of iterated polynomial mappings, Comm. Math. Phys. 88 (1983) 503-529.
6. A. Bogatyrev, Effective computation of Chebyshev polynomials for several intervals, Math. USSR Sb. 190 (1999) 1571-1605.
7. A. Draux, Polynomes Orthogonaux Formels - Applications, Lecture Notes in Mathematics, Vol. 974, Springer, Berlin, 1983.
8. B. Fischer, F. Peherstorfer, Chebyshev approximation via polynomial mappings and the convergence behaviour of Krylov subspace methods, Electron. Trans. Numer. Anal. 12 (2001) 205-215.
9. C.F. Gauss, Omnem functionem algebraicam rationalem integram, Gauß Werke Bd. III (1876) 1-30.
10. J.S. Geronimo, W. Van Assche, Orthogonal polynomials with asymptotically periodic recurrence coefficients, J. Approx. Theory 46 (1986) 251-283.
11. J.S. Geronimo, W. Van Assche, Orthogonal polynomials on several intervals via a polynomial mapping, Trans. Amer. Math. Soc. 308 (2) (1988) 559-581.
12. Ya.L Geronimus, On the character of the solutions of the moment problem in the case of a limit-periodic associated fraction, Izv. Akad. Nauk SSSR Ser. Math. 5 (1941) 203-210.
13. Ya.L Geronimus, On some finite difference equations and corresponding systems of orthogonal polynomials, Mem. Math. Sect. Fac. Phys. Kharkov State Univ. Kharkov Math. Soc. 25 (1975) 81-100.
14. S.O. Kamo, P.A. Borodin, Chebyshev polynomials for Julia sets, Moscow Univ. Math. Bull. 49 (1994) 44-45.
15. F. Marcellán, J. Petronilho, Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings ,Linear Algebra Appl. 260 (1997) 169-208.
16. F. Marcellán, J. Petronilho, Orthogonal polynomials and cubic polynomial mappings, Comm. Anal. Theory Contin. Fract. 8 (2000) 88-116.
17. F. Marcellán, G. Sansigre, Orthogonal polynomials and cubic transformations, J. Comput. Appl. Math. 49 (1993) 161-168.
18. P.B. Naiman, On the theory of periodic and limit-periodic Jacobian matrices, Soviet Math. Dokl. 3 (1962) 383-385.
19. A. Ostrowski, Über den ersten und vierten Gaußschen Beweis des Fundamentalsatzes der Algebra, Gauß Werke Bd. X 2, Abh. 3 (1933) 1-18.
20. F. Peherstorfer, On Tchebycheff polynomials on disjoint intervals, in: J. Szabados, K. Tandori, (Eds.), Haar Memorial Conference, Vol. II, North-Holland, Amsterdam, 1987, pp. 737-751.
21. F. Peherstorfer, On Bernstein-Szegö orthogonal polynomials on several intervals II: Orthogonal polynomials with periodic recurrence coefficients, J. Approx. Theory 64 (1991) 123-161.
22. F. Peherstorfer, Orthogonal and extremal polynomials on several intervals, J. Comput. Appl. Math. 48 (1993) 187-205.
23. F. Peherstorfer, Minimal polynomials for compact sets of the complex plane, Constr. Approx. 12 (1996) 481-488.
24. F. Peherstorfer, Deformation of minimal polynomials and approximation of several intervals by an inverse polynomial mapping, J. Approx. Theory 111 (2001) 180-195.
25. F. Peherstorfer, K. Schiefermayr, Description of Chebyshev polynomials on two arcs with the help of Jacobian theta functions, manuscript.
26. q-extremal polynomials on inverse images of polynomial mappings, J. Comput. Appl. Math. 127 (2001) 297-315.
27. V. Totik, Asymptotics for Christoffel functions, J. Analyse Math. 81 (2000) 283-303.
28. V. Totik, Polynomial inverse images and polynomial inequalities, Acta. Math. 187 (2001) 139-160.