Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

FULL QUADRATURE SUMS FOR GENERALIZED POLYNOMIALS WITH FREUD WEIGHTS

  • Published : 2010.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

Generalized nonnegative polynomials are defined as products of nonnegative polynomials raised to positive real powers. The generalized degree can be defined in a natural way. In this paper we extend quadrature sums involving pth powers of polynomials to those for generalized polynomials.

Keywords

References

  1. R. Askey, R. DeVore, G. Freud, J. Musielak, J. Peetre, and T. Popoviciu, Proposed problems, Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory) (Budapest, 1969), pp. 533-538. Akademiai Kiado, Budapest, 1972.
  2. T. Erdelyi, Bernstein and Markov type inequalities for generalized nonnegative polynomials, Canad. J. Math. 43 (1991), no. 3, 495-505. https://doi.org/10.4153/CJM-1991-030-3
  3. T. Erdelyi, Remez-type inequalities on the size of generalized polynomials, J. London Math. Soc. (2) 45 (1992), no. 2, 255-264. https://doi.org/10.1112/jlms/s2-45.2.255
  4. T. Erdelyi, A. Mate, and P. Nevai, Inequalities for generalized nonnegative polynomials, Constr. Approx. 8 (1992), no. 2, 241-255. https://doi.org/10.1007/BF01238273
  5. T. Erdelyi and P. Nevai, Generalized Jacobi weights, Christoffel functions, and zeros of orthogonal polynomials, J. Approx. Theory 69 (1992), no. 2, 111-132. https://doi.org/10.1016/0021-9045(92)90136-C
  6. G. Freud, On Markov-Bernstein-type inequalities and their applications, J. Approximation Theory 19 (1977), no. 1, 22-37. https://doi.org/10.1016/0021-9045(77)90026-0
  7. H. Joung, Estimates of Christoffel functions for generalized polynomials with exponential weights, Commun. Korean Math. Soc. 14 (1999), no. 1, 121-134.
  8. A. L. Levin and D. S. Lubinsky, Canonical products and the weights exp$(-{\left|}x{\right|}^{\alpha})$, ${\alpha}\;>\;1$ with applications, J. Approx. Theory 49 (1987), no. 2, 149-169. https://doi.org/10.1016/0021-9045(87)90085-2
  9. A. L. Levin and D. S. Lubinsky, Christoffel functions, orthogonal polynomials, and Nevai’s conjecture for Freud weights, Constr. Approx. 8 (1992), no. 4, 463-535. https://doi.org/10.1007/BF01203463
  10. D. S. Lubinsky, A. Mate, and P. Nevai, Quadrature sums involving pth powers of polynomials, SIAM J. Math. Anal. 18 (1987), no. 2, 531-544. https://doi.org/10.1137/0518041
  11. D. S. Lubinsky, H. N. Mhaskar, and E. B. Saff, A proof of Freud’s conjecture for exponential weights, Constr. Approx. 4 (1988), no. 1, 65-83. https://doi.org/10.1007/BF02075448
  12. D. S. Lubinsky and E. B. Saff, Strong Asymptotics for Extremal Polynomials Associated with Weights on R, Lecture Notes in Mathematics, 1305. Springer-Verlag, Berlin, 1988.
  13. D. S. Lubinsky and D. M. Matjila, Full quadrature sums for pth powers of polynomials with Freud weights, J. Comput. Appl. Math. 60 (1995), no. 3, 285-296. https://doi.org/10.1016/0377-0427(94)00045-3
  14. H. N. Mhaskar and E. B. Saff, Extremal problems for polynomials with exponential weights, Trans. Amer. Math. Soc. 285 (1984), no. 1, 203-234. https://doi.org/10.2307/1999480
  15. P. Nevai, Orthogonal polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, v+185 pp.
  16. Y. Xu, Mean convergence of generalized Jacobi series and interpolating polynomials. II, J. Approx. Theory 76 (1994), no. 1, 77-92. https://doi.org/10.1006/jath.1994.1006