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.
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
.
DOI: 10.1007/s10473-023-0108-5
[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
