zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Convergence and divergence of higher-order Hermite or Hermite-Fejér interpolation polynomials with exponential-type weights. (English) Zbl 1241.41002
Summary: Let $\Bbb R = (-\infty, \infty)$, and let $w_\rho(x) = |x|^\rho e^{-Q(x)}$, where $\rho > -1/2$ and $Q \in C^1(\Bbb R) : \Bbb R \rightarrow \Bbb R^+ = [0, \infty)$ 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:
41A05Interpolation (approximations and expansions)
42C05General theory of orthogonal functions and polynomials
WorldCat.org
Full Text: DOI
References:
[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 · doi:10.2748/tmj/1178225757
[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 · doi:10.1006/jath.1995.1024
[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 · doi:10.1016/0021-9045(92)90062-S
[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 · doi:10.1006/jath.1999.3371
[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 · doi:10.1016/S0021-9045(02)00041-2
[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 · doi:10.1016/j.jat.2004.01.006
[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 · doi:10.4134/CKMS.2009.24.2.303
[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 · doi:10.1016/j.jat.2010.02.006
[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 · doi:10.1007/BF01903540
[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 · doi:10.1007/BF00052097
[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 · doi:10.1155/2010/816363 · eudml:226231
[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 · doi:10.1016/j.jat.2007.12.004
[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 · doi:10.1155/2009/528454 · eudml:117811
[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 · emis:journals/SAT/papers/8/index.html · eudml:225908
[22] T. J. Rivlin, An Introduction to the Approximation of Functions, Dover, New York, NY, USA, 1981. · Zbl 0489.41001