OPTIMAL BIRKHOFF INTERPOLATION AND BIRKHOFF NUMBERS IN SOME FUNCTION SPACES*

doi:10.1007/s10473-023-0108-5

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (1): 125-142.doi: 10.1007/s10473-023-0108-5

OPTIMAL BIRKHOFF INTERPOLATION AND BIRKHOFF NUMBERS IN SOME FUNCTION SPACES^*

Guiqiao XU^1,†, Yongping Liu², Dandan GUO²

1. Department of Mathematics, Tianjin Normal University, Tianjin 300387, China;
2. Department of Mathematics, Beijing Normal University, Beijing 100875, China

收稿日期:2021-08-05 修回日期:2022-06-25 发布日期:2023-03-01
通讯作者: †Guiqiao XU.E-mail: Xuguiqiao@aliyun.com
基金资助:
*National Natural Science Foundation of China (11871006, 11671271).

OPTIMAL BIRKHOFF INTERPOLATION AND BIRKHOFF NUMBERS IN SOME FUNCTION SPACES^*

Guiqiao XU^1,†, Yongping Liu², Dandan GUO²

1. Department of Mathematics, Tianjin Normal University, Tianjin 300387, China;
2. Department of Mathematics, Beijing Normal University, Beijing 100875, China

Received:2021-08-05 Revised:2022-06-25 Published:2023-03-01
Contact: †Guiqiao XU.E-mail: Xuguiqiao@aliyun.com
About author:Yongping Liu, E-mail: ypliu@bnu.edu.cn; Dandan GUO, 2728561580@qq.com
Supported by:
*National Natural Science Foundation of China (11871006, 11671271).

摘要/Abstract

摘要： This paper investigates the optimal Birkhoff interpolation and Birkhoff numbers of some function spaces in space $L_\infty[-1,1]$ and weighted spaces $L_{p,\omega}[-1,1], \ 1\le p< \infty$, with $\omega$ being a continuous integrable weight function in $(-1,1)$. We proved that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal. We also show that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal when the function values of the two endpoints are included in the interpolation systems.

关键词: optimal Birkhoff interpolation, Birkhoff number, Sobolev space, worst case setting

Abstract: This paper investigates the optimal Birkhoff interpolation and Birkhoff numbers of some function spaces in space $L_\infty[-1,1]$ and weighted spaces $L_{p,\omega}[-1,1], \ 1\le p< \infty$, with $\omega$ being a continuous integrable weight function in $(-1,1)$. We proved that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal. We also show that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal when the function values of the two endpoints are included in the interpolation systems.

Key words: optimal Birkhoff interpolation, Birkhoff number, Sobolev space, worst case setting

Guiqiao XU, Yongping Liu, Dandan GUO. OPTIMAL BIRKHOFF INTERPOLATION AND BIRKHOFF NUMBERS IN SOME FUNCTION SPACES^*[J]. 数学物理学报(英文版), 2023, 43(1): 125-142.

Guiqiao XU, Yongping Liu, Dandan GUO. OPTIMAL BIRKHOFF INTERPOLATION AND BIRKHOFF NUMBERS IN SOME FUNCTION SPACES^*[J]. Acta mathematica scientia,Series B, 2023, 43(1): 125-142.

参考文献

[1] Wang H P, Xu G Q.Sampling numbers of a class of infinitely differentiable functions. J Math Anal App, 2020, 484: 123689
[2] Xu G Q, Wang H.Sample numbers and optimal Lagrange interpolation of Sobolev spaces. Rocky MT J Math, 2021, 51(1): 347-361
[3] Ben A, Yi S.Compressive Hermite Interpolation: Sparse, High-Dimensional Approximation from Gradient- Augmented Measurements. Constr Approx, 2019, 50: 167-207
[4] Dell’Accioa F, Tommaso F Di. Complete Hermite-Birkhoff interpolation on scattered data by combined Shepard operators. J Comput Appl Math, 2016, 300: 192-206
[5] Dell’Accioa F, Tommaso F Di, Nouisser O, Zerroudi B. Fast and accurate scattered Hermite interpolation by triangular Shepard operators. J Comput Appl Math, 2021, 382: 113092
[6] Garcáa-Marco I, Koiran P.Lower bounds by Birkhoff interpolation. J Complexity, 2017, 39: 38-50
[7] Goldman G.A case of multivariate Birkhoff interpolation using high order derivatives. J Approx Theory, 2017, 223: 19-28
[8] Jiao Y J, Wang L L, Huang C.Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis. J Comput Phys, 2016, 305: 1-28
[9] Mahmoodi A, Nazarzadeh A.A class of Birkhoff type interpolation and applications. Results Math, 2018, 73: 43
[10] Zare F, Heydari M, Loghmani G B, Wazwaz A-M.Numerical investigation of the Beam-type nanoelectrostatic actuator model by using the Birkhoff interpolation method. Int J Appl Comput Math, 2017, 3(Suppl 1): S129-S146
[11] Allasia G, Cavoretto R, De Rossi A.Hermite-Birkhoff interpolation on scattered data on the sphere and other manifolds. Appl Math Comput, 2018, 318: 35-50
[12] Barthelmann V, Novak E, Ritter K.High dimensional polynomial interpolation on sparse grids. Adv Comput Math, 2000, 12: 273-288
[13] ErrachidM, Essanhaji A,Messaoudi A. RMVPIA: a new algorithm for computing the Lagrange multivariate polynomial interpolation. Numer Math, 2020, 84(4): 1507-1534
[14] Hinrichs A, Novak E, Ullrich M.On weak tractability of the Clenshaw-Curtis Smolyak algorithm. J Approx Theory, 2014, 183: 31-44
[15] Irigoyen A.Multidimensional intertwining Leja sequences and applications in bidimensional Lagrange interpolation. J Approx Theory, 2021, 264: 105540
[16] Xu G Q.On weak tractability of the Smolyak algorithm for approximation problems. J Approx Theory, 2015, 192: 347-361
[17] Wilson L, Vaughn N, Krasny R.A GPU-accelerated fast multipole method based on barycentric Lagrange interpolation and dual tree traversal. Comput Phys Commun, 2021, 265: 108017
[18] Liu J, Zhu L Y.Bivariate Lagrange interpolation based on Chebyshev points of the second kind. Acta Math Hung, 2019, 159(2): 618-637
[19] Hoang N S.On node distributions for interpolation and spectral methods. Math Comp, 2016, 85: 667-692
[20] Babaev S S, Hayotov A R.Optimal interpolation formulas in W (^m,m-1) 2 space. Calcolo, 2019, 56: 23-45
[21] Xu G Q, Liu Z H, Wang H.Sample numbers and optimal Lagrange interpolation of Sobolev spaces Wr 1 . Chinese Ann Math, Ser B, 2021, 42(4): 519-528
[22] Lorentz G G, Jetter K, Riemenschneider S D.Birkhoff interpolation//Encyclopedia of Mathematics and its Applications, Vol 19. Cambridge University Press, 1984
[23] Liu Z H, Lu W T, Xu G Q.Simultaneous approximation of Birkhoff interpolation and the associated sharp inequalities. Int J Wavelets Multi, 2020, 18(4): 2050021
[24] Nürnberger G.Approximation by Spline Functions. Beijing: Springer-Verlag, 1992
[25] Novak E, Woźniakowski H.Tractability of Multivariate Problems. Volume I: Linear Information. Zürich: Eur Math Soc, 2008
[26] DeVore R A, Lorentz G G. Constructive Approximation. New York: Springer-Verlag, 1993

OPTIMAL BIRKHOFF INTERPOLATION AND BIRKHOFF NUMBERS IN SOME FUNCTION SPACES^*

OPTIMAL BIRKHOFF INTERPOLATION AND BIRKHOFF NUMBERS IN SOME FUNCTION SPACES^*

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	李思然. A NEW PROOF OF GAFFNEY’S INEQUALITY FOR DIFFERENTIAL FORMS ON MANIFOLDS-WITH-BOUNDARY: THE VARIATIONAL APPROACH À LA KOZONO–YANAGISAWA[J]. 数学物理学报(英文版), 2022, 42(4): 1427-1452.
[2]	邓方文, 欧阳才衡, 彭茹. SOME QUESTIONS REGARDING VERIFICATION OF CARLESON MEASURES[J]. 数学物理学报(英文版), 2021, 41(6): 2136-2148.
[3]	李丹, 李俊峰, 肖杰. AN UPBOUND OF HAUSDORFF'S DIMENSION OF THE DIVERGENCE SET OF THE FRACTIONAL SCHRÖDINGER OPERATOR ON H^s($\mathbb{R}^n$)[J]. 数学物理学报(英文版), 2021, 41(4): 1223-1249.
[4]	Abdelkrim MOUSSAOUI, Jean VELIN. EXISTENCE AND BOUNDEDNESS OF SOLUTIONS FOR SYSTEMS OF QUASILINEAR ELLIPTIC EQUATIONS[J]. 数学物理学报(英文版), 2021, 41(2): 397-412.
[5]	郭常予, 向长林. REGULARITY OF P-HARMONIC MAPPINGS INTO NPC SPACES[J]. 数学物理学报(英文版), 2021, 41(2): 633-645.
[6]	戴济能, 周静云. GLEASON'S PROBLEM ON FOCK-SOBOLEV SPACES[J]. 数学物理学报(英文版), 2021, 41(1): 337-348.
[7]	Jae-Myoung KIM, Yun-Ho KIM, Jongrak LEE. RADIALLY SYMMETRIC SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING NONHOMOGENEOUS OPERATORS IN AN ORLICZ-SOBOLEV SPACE SETTING[J]. 数学物理学报(英文版), 2020, 40(6): 1679-1699.
[8]	陶文宇, 陈艳萍, 李纪黎. GRADIENT ESTIMATES FOR THE COMMUTATOR WITH FRACTIONAL DIFFERENTIATION FOR SECOND ORDER ELLIPTIC OPERATORS[J]. 数学物理学报(英文版), 2019, 39(5): 1255-1264.
[9]	程峰, 李维喜, 徐超江. GEVREY REGULARITY WITH WEIGHT FOR INCOMPRESSIBLE EULER EQUATION IN THE HALF PLANE[J]. 数学物理学报(英文版), 2017, 37(4): 1115-1132.
[10]	肖杰胜, 曹广福. WANDERING SUBSPACES OF THE HARDY-SOBOLEV SPACES OVER Dⁿ[J]. 数学物理学报(英文版), 2016, 36(5): 1467-1473.
[11]	吴云顺, 谭忠. ASYMPTOTIC BEHAVIOR OF THE STOKES APPROXIMATION EQUATIONS FOR COMPRESSIBLE FLOWS IN R³[J]. 数学物理学报(英文版), 2015, 35(3): 746-760.
[12]	徐艳艳, 陈广贵, 甘莹, 许燕. PROBABILISTIC AND AVERAGE LINEAR WIDTHS OF SOBOLEV SPACE WITH GAUSSIAN MEASURE IN SPACE S_Q(T) (1≤Q≤∞)[J]. 数学物理学报(英文版), 2015, 35(2): 495-507.
[13]	傅仰耿, 刘正荣, 唐昊. NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE[J]. 数学物理学报(英文版), 2014, 34(6): 1781-1794.
[14]	王宏伟. LOCAL WELL-POSEDNESS IN SOBOLEV SPACES WITH NEGATIVE INDICES FOR A SEVENTH ORDER DISPERSIVE EQUATION[J]. 数学物理学报(英文版), 2014, 34(1): 199-208.
[15]	彭茹, 欧阳才衡. CARLESON MEASURES FOR BESOV-SOBOLEV SPACES WITH APPLICATIONS IN THE UNIT BALL OF Cⁿ[J]. 数学物理学报(英文版), 2013, 33(5): 1219-1230.