It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same.
The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss recent uniform estimates in comonotone approximation, mention recent developments and state several open problems in the (co)convex case, and reiterate that co--monotone approximation with is completely different from comonotone and coconvex cases.
Additionally, we show that, for each function from , the set of all monotone functions on , and every , we have
where denotes the set of algebraic polynomials of degree , , and depends only on .
Keywords: Approximation by algebraic polynomials, shape preserving approximation, constrained approximation
@article{SMAI-JCM_2019__S5__99_0, author = {Kopotun, K. A. and Leviatan, D. and Shevchuk, I. A.}, title = {Uniform and pointwise shape preserving approximation {(SPA)} by algebraic polynomials: an update}, journal = {The SMAI Journal of computational mathematics}, pages = {99--108}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {S5}, year = {2019}, doi = {10.5802/smai-jcm.54}, language = {en}, url = {http://www.numdam.org/articles/10.5802/smai-jcm.54/} }
TY - JOUR AU - Kopotun, K. A. AU - Leviatan, D. AU - Shevchuk, I. A. TI - Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update JO - The SMAI Journal of computational mathematics PY - 2019 SP - 99 EP - 108 VL - S5 PB - Société de Mathématiques Appliquées et Industrielles UR - http://www.numdam.org/articles/10.5802/smai-jcm.54/ DO - 10.5802/smai-jcm.54 LA - en ID - SMAI-JCM_2019__S5__99_0 ER -
%0 Journal Article %A Kopotun, K. A. %A Leviatan, D. %A Shevchuk, I. A. %T Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update %J The SMAI Journal of computational mathematics %D 2019 %P 99-108 %V S5 %I Société de Mathématiques Appliquées et Industrielles %U http://www.numdam.org/articles/10.5802/smai-jcm.54/ %R 10.5802/smai-jcm.54 %G en %F SMAI-JCM_2019__S5__99_0
Kopotun, K. A.; Leviatan, D.; Shevchuk, I. A. Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update. The SMAI Journal of computational mathematics, Volume S5 (2019), pp. 99-108. doi : 10.5802/smai-jcm.54. http://www.numdam.org/articles/10.5802/smai-jcm.54/
[1] On Jackson–Stechkin type estimates for piecewise q-convex approximation of functions, Visnyk. Math. Mech., Kyiv. Univ. Im. Tarasa Shevchenka, Volume 36 (2016), pp. 6-10
[2] Constructive approximation, Grundlehren der Mathematischen Wissenschaften, 303, Springer, 1993, x+449 pages | MR | Zbl
[3] Pointwise estimates for monotone polynomial approximation, Constr. Approx., Volume 1 (1985) no. 4, pp. 323-331 | DOI | MR | Zbl
[4] Theory of uniform approximation of functions by polynomials, Walter de Gruyter, 2008, xvi+480 pages
[5] Interpolatory pointwise estimates for polynomial approximation, Constr. Approx., Volume 16 (2000) no. 4, pp. 603-629 | DOI | MR | Zbl
[6] Uniform and pointwise shape preserving approximation by algebraic polynomials, Surv. Approx. Theory, Volume 6 (2011), pp. 24-74 | MR | Zbl
[7] Interpolatory pointwise estimates for monotone polynomial approximation, J. Math. Anal. Appl., Volume 459 (2018) no. 2, pp. 1260-1295 | DOI | MR | Zbl
[8] Interpolatory estimates for convex piecewise polynomial approximation, J. Math. Anal. Appl., Volume 474 (2019) no. 1, pp. 467-479 | DOI | MR | Zbl
[9] Pointwise estimates for convex polynomial approximation, Proc. Am. Math. Soc., Volume 98 (1986) no. 3, pp. 471-474 | DOI | MR | Zbl
[10] Positive results and counterexamples in comonotone approximation, Constr. Approx., Volume 36 (2012) no. 2, pp. 243-266 | DOI | MR | Zbl
[11] Counterexamples in convex and higher order constrained approximation, East J. Approx., Volume 1 (1995) no. 3, pp. 391-398 | MR | Zbl
[12] Comparing the degrees of unconstrained and shape preserving approximation by polynomials, J. Approximation Theory, Volume 211 (2016), pp. 16-28 | DOI | MR | Zbl
[13] Jackson type estimates for piecewise -monotone approximation, , are not valid, Pure Appl. Funct. Anal., Volume 1 (2016) no. 1, pp. 85-96 | Zbl
[14] Positive results and counterexamples in comonotone approximation II, J. Approximation Theory, Volume 179 (2014), pp. 1-23 | DOI | MR | Zbl
[15] Degree of approximation by monotone polynomials. II, J. Approximation Theory, Volume 2 (1969), pp. 265-269 | DOI | MR | Zbl
Cited by Sources: