Geodesic
An optimal interpolation problem with hermite information in the sobolev class $W^{n}_{1}([-1,1])$
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 1, pp. 270-279
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In this paper, we study the optimal interpolation problem in the Sobolev class $W^{n}_{1}([-1,1])$, $n\in\mathbb N$, with Hermite information. By some properties of spline functions, we proved that the Lagrange interpolation based on the extreme points of Chebyshev polynomials is optimal for $W^{n}_{1}([-1,1])$, and we obtained the approximation error for the optimal interpolation problem.
Keywords: spline function, Sobolev class.
Mots-clés : Hermite interpolation, optimal interpolation 
@article{TIMM_2024_30_1_a19,
     author = {Dandan Guo and Yongping Liu and Guiqiao Xu},
     title = {An optimal interpolation problem with hermite information in the sobolev class $W^{n}_{1}([-1,1])$},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {270--279},
     year = {2024},
     volume = {30},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a19/}
}
TY  - JOUR
AU  - Dandan Guo
AU  - Yongping Liu
AU  - Guiqiao Xu
TI  - An optimal interpolation problem with hermite information in the sobolev class $W^{n}_{1}([-1,1])$
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2024
SP  - 270
EP  - 279
VL  - 30
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a19/
LA  - en
ID  - TIMM_2024_30_1_a19
ER  - 
%0 Journal Article
%A Dandan Guo
%A Yongping Liu
%A Guiqiao Xu
%T An optimal interpolation problem with hermite information in the sobolev class $W^{n}_{1}([-1,1])$
%J Trudy Instituta matematiki i mehaniki
%D 2024
%P 270-279
%V 30
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a19/
%G en
%F TIMM_2024_30_1_a19
Dandan Guo; Yongping Liu; Guiqiao Xu. An optimal interpolation problem with hermite information in the sobolev class $W^{n}_{1}([-1,1])$. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 1, pp. 270-279. http://geodesic.mathdoc.fr/item/TIMM_2024_30_1_a19/

[1] Traub J.F.; Woźniakowsi H., A General Theory of optimal algorithms, Acad. Press, Inc., NY, USA, 1980 | MR | Zbl

[2] Sun Y.S., Fang G.S., Approximation theory of functions, v. II, Beijing Normal Univ Press, Beijing, China, 1990 (in Chinese)

[3] Temlyakov V., Multivariate approximation, Cambridge Univ Press, Cambridge, UK, 2018 | MR | Zbl

[4] Vybiral J., “Sampling numbers and function spaces”, J. Complexity, 23 (2007), 773–792 | DOI | MR | Zbl

[5] Wang H., Wang K., “Optimal recovery of Besov classes of generalized smoothness and Sobolev classes on the sphere”, J. Complexity, 32:1 (2016), 40–52 | DOI | MR | Zbl

[6] Wang H.P., Xu G.Q., “Sampling numbers of a class of infinitely differentiable functions”, J. Math. Anal. Appl., 484:1 (2020), 123689 | DOI | MR | Zbl

[7] Ben A.; Yi S., “Compressive Hermite interpolation: sparse, high-dimensional approximation from gradient-augmented measurements”, Constr. Approx., 50 (2019), 167–207 | DOI | MR | Zbl

[8] Peng J., Hampton J., Doostan A., “On polynomial chaos expansion via gradient-enhanced $\ell_1$ minimization”, J. Comput. Phys., 310 (2016), 440–458 | DOI | MR | Zbl

[9] Allasia G., Cavoretto R., De Rossi A., “Hermite–Birkhoff interpolation on scattered data on the sphere and other manifolds”, Appl. Math. Comput., 318 (2018), 35–50 | DOI | MR | Zbl

[10] Xie T.F., Zhou S.P., Approximation theory of real function, Hangzhou Univ Press, Hangzhou, China, 1998 (in Chinese)

[11] Babaev S.S., Hayotov A.R., “Optimal interpolation formulas in $W^{(m,m-1)}_{2}$ space”, Calcolo, 56 (2019), 23 | DOI | MR | Zbl

[12] Mastroianni G, Occorsio D., “Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey”, J. Comput. Appl. Math., 134 (2001), 325–341 | DOI | MR | Zbl

[13] Hoang N.S., “On node distributions for interpolation and spectral methods”, Math. Comput., 85 (2016), 667–692 | DOI | MR | Zbl

[14] Xu G.Q., Wang H., “Sample numbers and optimal Lagrange interpolation in Sobolev spaces”, Rocky MT. J. Math., 51:1 (2021), 347–361 | DOI | MR | Zbl

[15] Rack H.J., Vajda R., “On Optimal quadratic Lagrange interpolation: Extremal node systems with minimal Lebesgue constant via symbolic computation”, Serdica J. Comput., 8 (2014), 71–96 | DOI | MR | Zbl

[16] Rack H.J., Vajda R., “Optimal cubic Lagrange interpolation: Extremal node systems with minimal Lebesgue constant”, Studia Univ. Babes-Bolyai Math., 60:2 (2015), 151–171 | MR | Zbl

[17] Kofanov V.A., “Approximation in the mean of classes of differentiable functions by algebraic polynomials”, Math. USSR Izv., 23:2 (1984), 353–365 | DOI | MR | Zbl

[18] Kofanov V.A., “Exact values for best approximations in the mean of the classes $W^{r}_{L}$ $(r=1,2)$ by algebraic polynomials”, Studies in Contemporary Problems of Summation and Approximation of Functions and Their Applications, Gos. Univ., Dnepropetrovsk, Dnepropetrovsk, 1977, 18–20 (in Russian)

[19] Shevaldina O.Ya., “Approximation of the classes $W_{p}^{r}$ by polynomial splines in the mean”, Approximation in Specific and Abstract Banach Spaces, Akad. Nauk SSSR, Ural. Nauchn. Tsentr, Sverdlovsk, 1987, 113–120 (in Russian) | MR

[20] Xu G.Q., Liu Z.H., Wang H., “Sample numbers and optimal Lagrange Interpolation of Sobolev spaces $W_{1}^{r}$”, Chin. Ann. Math. Ser. B., 42:4 (2021), 519–528 | DOI | MR | Zbl

[21] Nürnberger, G., Approximation by Spline functions, Springer-Verlag, Berlin, 1989 | MR | Zbl