Geodesic
On the error of optimal interpolation by linear shape-preserving algorithms
Sibirskij žurnal industrialʹnoj matematiki, Tome 15 (2012) no. 2, pp. 119-127.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the problem of optimal linear interpolation by algorithms positive on a cone describing the properties of the shape of the functions being approximated. We show that such linear shape-preserving methods have a negative property connected with the inability to identically approximate algebraic polynomials of at least the given degree. We also show that the estimation of the error of the problem of linear shape-preserving interpolation can be reduced to the problem of conic optimization. This makes it possible to use the duality principle for obtaining an estimate of the error of the shape-preserving interpolation.
Mots-clés : optimal interpolation
Keywords: shape-preserving approximation, conic programming. 
@article{SJIM_2012_15_2_a11,
     author = {S. P. Sidorov},
     title = {On the error of optimal interpolation by linear shape-preserving algorithms},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {119--127},
     publisher = {mathdoc},
     volume = {15},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2012_15_2_a11/}
}
TY  - JOUR
AU  - S. P. Sidorov
TI  - On the error of optimal interpolation by linear shape-preserving algorithms
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2012
SP  - 119
EP  - 127
VL  - 15
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2012_15_2_a11/
LA  - ru
ID  - SJIM_2012_15_2_a11
ER  - 
%0 Journal Article
%A S. P. Sidorov
%T On the error of optimal interpolation by linear shape-preserving algorithms
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2012
%P 119-127
%V 15
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2012_15_2_a11/
%G ru
%F SJIM_2012_15_2_a11
S. P. Sidorov. On the error of optimal interpolation by linear shape-preserving algorithms. Sibirskij žurnal industrialʹnoj matematiki, Tome 15 (2012) no. 2, pp. 119-127. http://geodesic.mathdoc.fr/item/SJIM_2012_15_2_a11/

[1] Gal S. G., Shape-Preserving Approximation by Real and Complex Polynomials, Springer-Verl., Boston, 2008 | MR

[2] Leviatan D., “Shape-preserving approximation by polynomials”, J. Comput. Appl. Math., 121 (2000), 73–94 | DOI | MR | Zbl

[3] Micchelli C. A., Rivlin T. J., “Lectures on optimal recovery”, Lectures Notes in Math., 1129, 1985, 21–93 | DOI | MR

[4] Traub J. F., Wozniakowski H., A General Theory of Optimal Algorithms, Acad. Press, N.Y., 1980 | MR

[5] Muñoz-Delgado F. J., Ramírez-González V., “Almost convexity and quantitative Korovkin type results”, Appl. Math. Lett., 11:4 (1998), 105–108 | DOI | MR | Zbl

[6] Muñoz-Delgado F. J., Ramírez-González V., Cárdenas-Morales D., “Qualitative Korovkin-type results on conservative approximation”, J. Approx. Theory, 94 (1998), 144–159 | DOI | MR | Zbl

[7] Kalmykov M. U., Sidorov S. P., “A moment problem for discrete nonpositive measures on a finite interval”, Internat. J. Math. Math. Sci., 2011 (2011), article ID 545780, 8 | DOI | MR | Zbl

[8] Korovkin P. P., “O poryadke priblizheniya funktsii lineinymi polozhitelnymi operatorami”, Dokl. AN SSSR, 114:6 (1957), 1158–1161 | MR | Zbl

[9] Videnskii V. S., “Ob odnom tochnom neravenstve dlya lineinykh polozhitelnykh operatorov konechnogo ranga”, Dokl. AN TadzhSSR, 24:12 (1981), 715–717 | MR

[10] Magaril-Ilyaev G. G., “Printsip Lagranzha dlya gladkikh zadach s ogranicheniyami na konuse”, Vladikavkaz. mat. zhurn., 7:4 (2005), 38–45 | MR

[11] Shapiro A., “On duality theory of conic linear problems”, Semi-Infinite Programming: Recent Advances, Kluwer Acad. Publ., Dordrecht, 2002, 135–165 | MR

[12] Mupasiri D., Prophet M., “A note on the existence of shape-preserving projections”, Rocky Mountain J. Math., 37:2 (2007), 573–585 | DOI | MR | Zbl

[13] Mupasiri D., Prophet M. P., “On the difficulty of preserving monotonicity via projections and related results”, Jaen J. Approx., 2:1 (2010), 1–12 | MR

[14] Lewicki G., Prophet M., “Minimal shape-preserving projections in tensor product spaces”, J. Approx. Theory, 162:5 (2010), 931–951 | DOI | MR | Zbl

[15] Sidorov S., “Approximation of the rth differential operator by means of linear shape preserving opeartors of finite rank”, J. Approx. Theory, 124:2 (2003), 232–241 | DOI | MR | Zbl