Sample Numbers and Optimal Lagrange Interpolation of Sobolev Spaces W ₁ ^r

Abstract

This paper investigates the optimal recovery of Sobolev spaces W ₁ ^r[−1, 1], r ∈ ℕ in the space L ₁[−1, 1]. They obtain the values of the sampling numbers of W ₁ ^r[−1, 1] in L ₁[−1, 1] and show that the Lagrange interpolation algorithms based on the extreme points of Chebyshev polynomials are optimal algorithms. Meanwhile, they prove that the extreme points of Chebyshev polynomials are optimal Lagrange interpolation nodes.

Keywords

Worst case setting / Sampling number / Optimal Lagrange interpolation nodes / Sobolev space

References

[1] Byrenheid G, Dũng D, Sickel W, Ullrich T. Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in H^γ. J. Approx. Theory, 2016, 207: 207-231

[2] Byrenheid G, Kämmerer L, Ullrich T, Volkmer T. Tight error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothness. Numer. Math., 2017, 136(4): 993-1034

[3] Byrenheid G, Ullrich T. Optimal sampling recovery of mixed order Sobolev embeddings via discrete Littlewood-Paley type characterizations. Anal. Mathem., 2017, 43(2): 133-191

[4] Dũng D. B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness. J. Complexity, 2011, 27: 541-567

[5] Dũng D. Sampling and cubature on sparse grids based on a B-spline quasi-interpolation. Found. Comp. Math., 2016, 16: 1193-1240

[6] Dũng D. B-spline quasi-interpolation sampling representation and sampling recovery in Sobolev spaces of mixed smoothness. Acta Math. Vietnamica, 2018, 43: 83-110

[7] Dũng D. Optimal adaptive sampling recovery. Adv. Comput. Math., 2011, 34: 1-41

[8] Dũng D, Temlyakov V, Ullrich T. Hyperbolic Cross Approximation, Advanced Courses in Mathematics CRM Barcelona, 2018, Barecelona: Springer Nature Switzerland AG

[9] Kudryavtsev S N. The best accuracy of reconstruction of finitely smooth functions from their values at a given number of points. Izv. Math., 1998, 62(1): 19-53

[10] Novak E, Triebel H. Function spaces in Lipschitz domains and optimal rates of convergence for sampling. Constr. Approx., 2006, 23: 325-350

[11] Temlyakov V. Multivariate Approximation, 2018, Cambridge: Cambridge University Press

[12] Triebel H. Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration, 2010, Zürich: EMS

[13] Vybíral J. Sampling numbers and function spaces. J. Complexity, 2007, 23: 773-792

[14] Wang H, Wang K. Optimal recovery of Besov classes of generalized smoothness and Sobolev classes on the sphere. J. Complexity, 2016, 32: 40-52

[15] Xu G Q, Wang H. Sample numbers and optimal Lagrange interpolation of Sobolev spaces. Rocky MT. J. Math., 2021, 51(1): 347-361

[16] Babaev S S, Hayotov A R. Optimal interpolation formulas in W ₂ ^m,m−1 space. Calcolo, 2019, 56: 23-45

[17] Mastroianni G, Occorsio D. Optimal systems of nodes for Lagrange interpolation on bounded intervals, A survey. J. Comput. Appl. Math., 2001, 134: 325-341

[18] Hoang N S. On node distributions for interpolation and spectral methods. Math. Comp., 2016, 85: 667-692

[19] Szabados J, Vèrtesi P. Interpolation of Functions, 1990, Singapore: World Scientific

[20] Rack H J, Vajda R. On optimal quadratic Lagrange interpolation: Extremal node systems with minimal Lebesgue constant via symbolic computation. Serdica J. Comput., 2014, 8: 71-96

[21] Rack H J, Vajda R. Optimal cubic Lagrange interpolation: Extremal node systems with minimal Lebesgue constant. Stud. Univ. Babes-Bolyai Math., 2015, 60(2): 151-171

[22] Birkhoff G D. General mean value and remainder theorems with applications to mechanical differentiation and quadrature. Trans. Am. Math. Soc., 1906, 7: 107-136

[23] Nürnberger G. Approximation by Spline Functions, 1992, Beijing: Springer-Verlag

[24] Novak E, Woźniakowski H. Tractability of Multivariate Problems, Volume I: Linear Information, 2008, Zürich: EMS

[25] Devore R A, Lorentz G G. Constructive Approximation, 1993, New York: Springer-Verlag