In this paper, the rational interpolation of |x| at the extended Chebyshev nodes was studied. That the exact order of approximation is O(1/(nln n))was obtained. Through numerical calculation, it is found that the error of the same approximation order is related to the density of nodes and the concavity and convexity of the curve where the nodes are located.
LI Jianjun
,
ZHANG Huiming
. On rational interpolation to |x| at the extended Chebyshev nodes[J]. Journal of Hebei University(Natural Science Edition), 2023
, 43(1)
: 16
-20
.
DOI: 10.3969/j.issn.1000-1565.2023.01.003
[1] BERNSTEIN S. Sur la meilleure approximation de |x| par des polynomes de degrés donnés[J]. Acta Math, 1914, 37(1): 1-57. DOI: 10.1007/BF02401828.
[2] NEWMAN D J. Rational approximation to |x|[J]. Michigan Math J, 1964, 11(1):11-14. DOI: 10.1307/mmj/1028999029.
[3] BRUTMAN L, PASSOW E. On rational interpolation to |x|[J]. Constr Approx, 1997, 13(3): 381-391. DOI: 10.1007/s003659900049.
[4] WERNER H. Rationale interpolation von |x| in ä quidistanten Punkten[J]. Math Z, 1982, 180(1): 11-17. DOI: 10.1007/BF01214996.
[5] BRUTMAN L, PASSOW E. Rational interpolation to |x| at the Chebyshev nodes[J]. Bull Austral Math Soc, 1997, 56(1): 81-86. DOI: 10.1017/s0004972700030756.
[6] 张慧明,李建俊.|x|α在Chebyshev结点的有理插值[J].黑龙江大学自然科学学报, 2016, 33(4): 491-495. DOI: 10.13482/j.issn1001-7011.2015.09.261.
[7] 张慧明,段生贵,李建俊.|x|α在第二类Chebyshev结点的有理插值[J].四川师范大学学报(自然科学版), 2015, 38(6): 889-892. DOI: 10.3969/j.issn.1001-8395.2015.06.019.
[8] 张慧明,段生贵,李建俊.|x|α(1≤α<2)在等距结点的有理插值[J].华中师范大学学报(自然科学版), 2016, 50(1): 21-23, 27. DOI: 10.19603/j.cnki.1000-1190.2016.01.005.
[9] BRUTMAN L. On rational interpolation to |x| at the adjusted Chebyshev nodes[J]. J Approx Theory, 1998, 95(1): 146-152. DOI: 10.1006/jath.1998.3206.
[10] 张慧明,李建俊.|x|在第二类Chebyshev结点的有理逼近[J].郑州大学学报(理学版), 2010, 42(2): 1-3, 9.
[11] 张慧明,门玉梅,李建俊.|x|在正切结点组的有理插值[J].天津师范大学学报(自然科学版), 2011, 31(4): 5-6. DOI: 10.3969/j.issn.1671-1114.2011.04.002.
[12] 张慧明,李建俊,段继光.|x|在调整的第二类Chebyshev结点组的有理插值[J].数学杂志, 2014, 34(3): 509-514. DOI: 10.13548/j.sxzz.2014.03.012.
[13] 张慧明,段生贵,李建俊,等. |x|的有理插值[J].高等学校计算数学学报, 2016, 38(1): 52-59.
[14] HAN X L. On the order of approximation for the rational interpolation to |x|[J]. Approx Theory & its Appl, 2002, 18(2): 58-64. DOI: 10.1007/BF02837401.
[15] XIE T F, ZHOU S P. The asymptotic property of approximation to |x| by Newman’s rational operators[J].Acta Math Hung, 2004, 103(4): 313-319. DOI: 10.1023/B: AMHU.0000028831.89374.d3.
[16] XIE T F, ZHOU X L. Improvement of Newman inequality[J].J Math Anal Appl, 2006, 315(1): 359-366. DOI: 10.1016/j.jmaa.2005.06.066.
[17] ZHAN Q, XU S S, ZHANG Y H. Asymptotic property of approximation to xαsgnx by Newman type operators[J].Acta Math Appl Sin Engl Ser, 2010, 26(4): 617-624. DOI: 10.1007/s10255-007-7147-x.
[18] 张慧明, 李建俊. |x|在加密Newman结点的有理插值[J]. 工程数学学报, 2018, 35(4): 408-414. DOI: 10.3969/j.issn.1005-3085.2018.04.004.
[19] 程一元,张永全,查星星.|x|α(1≤α<2)在调整的正切节点组的有理逼近[J].工程数学学报, 2019, 36(5): 551-556. DOI: 10.3969/j.issn.1005-3085.2019.05.006. (
