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Convergence and divergence of higher-order Hermite or Hermite-Fejér interpolation polynomials with exponential-type weights. (English) Zbl 1241.41002

Summary: Let \(\mathbb R = (-\infty, \infty)\), and let \(w_\rho(x) = |x|^\rho e^{-Q(x)}\), where \(\rho > -1/2\) and \(Q \in C^1(\mathbb R) : \mathbb R \rightarrow \mathbb R^+ = [0, \infty)\) is an even function. Then we can construct the orthonormal polynomials \(p_n(w^2_\rho; x)\) of degree \(n\) for \(w^2_\rho(x)\). In this paper for an even integer \(v \geq 2\) we investigate the convergence theorems with respect to the higher-order Hermite and Hermite-Fejér interpolation polynomials and related approximation process based at the zeros \(\{x_{k,n,\rho}\}^n_{k=1}\) of \(p_n(w^2_{\rho}; x)\). Moreover, for an odd integer \(v \geq 1\), we give a certain divergence theorem with respect to the higher-order Hermite-Fejér interpolation polynomials based at the zeros \(\{x_{k,n,\rho}\}^n_{k=1}\) of \(p_n(w^2_\rho; x)\).

MSC:

41A05 Interpolation in approximation theory
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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[1] Y. Kanjin and R. Sakai, “Pointwise convergence of Hermite-Fejér interpolation of higher order for Freud weights,” The Tohoku Mathematical Journal, vol. 46, no. 2, pp. 181-206, 1994. · Zbl 0807.41004
[2] Y. Kanjin and R. Sakai, “Convergence of the derivatives of Hermite-Fejér interpolation polynomials of higher order based at the zeros of Freud polynomials,” Journal of Approximation Theory, vol. 80, no. 3, pp. 378-389, 1995. · Zbl 0819.41002
[3] D. S. Lubinsky, “Hermite and Hermite-Fejér interpolation and associated product integration rules on the real line: the L1 theory,” Journal of Approximation Theory, vol. 70, no. 3, pp. 284-334, 1992. · Zbl 0777.41015
[4] T. Kasuga and R. Sakai, “Uniform or mean convergence of Hermite-Fejér interpolation of higher order for Freud weights,” Journal of Approximation Theory, vol. 101, no. 2, pp. 330-358, 1999. · Zbl 0946.41003
[5] T. Kasuga and R. Sakai, “Orthonormal polynomials with generalized Freud-type weights,” Journal of Approximation Theory, vol. 121, no. 1, pp. 13-53, 2003. · Zbl 1034.42021
[6] T. Kasuga and R. Sakai, “Orthonormal polynomials for generalized Freud-type weights and higher-order Hermite-Fejér interpolation polynomials,” Journal of Approximation Theory, vol. 127, no. 1, pp. 1-38, 2004. · Zbl 1053.42026
[7] T. Kasuga and R. Sakai, “Orthonormal polynomials for Laguerre-type weights,” Far East Journal of Mathematical Sciences, vol. 15, no. 1, pp. 95-105, 2004. · Zbl 1083.42020
[8] T. Kasuga and R. Sakai, “Conditions for uniform or mean convergence of higher order Hermite-Fejér interpolation polynomials with generalized Freud-type weights,” Far East Journal of Mathematical Sciences, vol. 19, no. 2, pp. 145-199, 2005. · Zbl 1119.41002
[9] A. L. Levin and D. S. Lubinsky, Orthogonal Polynomials for Exponential Weights, Springer, New York, NY, USA, 2001. · Zbl 0997.42011
[10] H. S. Jung and R. Sakai, “Specific examples of exponential weights,” Korean Mathematical Society, vol. 24, no. 2, pp. 303-319, 2009. · Zbl 1168.41305
[11] H. S. Jung and R. Sakai, “The Markov-Bernstein inequality and Hermite-Fejér interpolation for exponential-type weights,” Journal of Approximation Theory, vol. 162, no. 7, pp. 1381-1397, 2010. · Zbl 1195.41003
[12] R. Sakai, “Hermite-Fejér interpolation,” in Approximation Theory, vol. 58, pp. 591-601, North-Holland, Amsterdam, The Netherlands, 1990. · Zbl 0817.41005
[13] R. Sakai, “The degree of approximation of differentiable functions by Hermite interpolation polynomials,” Acta Mathematica Hungarica, vol. 58, no. 1-2, pp. 9-11, 1991. · Zbl 0748.41012
[14] R. Sakai, “Hermite-Fejér interpolation prescribing higher order derivatives,” in Progress in Approximation Theory, pp. 731-759, Academic Press, Boston, Mass, USA, 1991.
[15] R. Sakai, “Certain bounded Hermite-Fejer interpolation polynomials operator,” Acta Mathematica Hungarica, vol. 59, pp. 111-114, 1992. · Zbl 0777.41001
[16] R. Sakai and P. Vértesi, “Hermite-Fejér interpolations of higher order. III,” Studia Scientiarum Mathematicarum Hungarica, vol. 28, no. 1-2, pp. 87-97, 1993. · Zbl 0802.41006
[17] R. Sakai and P. Vértesi, “Hermite-Fejér interpolations of higher order. IV,” Studia Scientiarum Mathematicarum Hungarica, vol. 28, no. 3-4, pp. 379-386, 1993. · Zbl 0832.41003
[18] H. S. Jung and R. Sakai, “Derivatives of orthonormal polynomials and coefficients of Hermite-Fejér interpolation polynomials with exponential-type weights,” Journal of Inequalities and Applications, vol. 2010, Article ID 816363, 29 pages, 2010. · Zbl 1194.33010
[19] H. S. Jung and R. Sakai, “Orthonormal polynomials with exponential-type weights,” Journal of Approximation Theory, vol. 152, no. 2, pp. 215-238, 2008. · Zbl 1152.41001
[20] H. S. Jung and R. Sakai, “Derivatives of integrating functions for orthonormal polynomials with exponential-type weights,” Journal of Inequalities and Applications, vol. 2009, Article ID 528454, 22 pages, 2009. · Zbl 1176.41011
[21] D. S. Lubinsky, “A survey of weighted polynomial approximation with exponential weights,” Surveys in Approximation Theory, vol. 3, pp. 1-105, 2007. · Zbl 1181.41004
[22] T. J. Rivlin, An Introduction to the Approximation of Functions, Dover, New York, NY, USA, 1981. · Zbl 0489.41001
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