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The degree of coconvex polynomial approximation

Authors: K. Kopotun, D. Leviatan and I. A. Shevchuk
Journal: Proc. Amer. Math. Soc. 127 (1999), 409-415
MSC (1991): Primary 41A10, 41A17, 41A25, 41A29
MathSciNet review: 1459130
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Abstract: Let $f\in C[-1,1]$ change its convexity finitely many times in the interval, say $s$ times, at $Y_{s}:-1<y_{1}<\cdots <y_{s}<1$. We estimate the degree of approximation of $f$ by polynomials of degree $n$, which change convexity exactly at the points $Y_{s}$. We show that provided $n$ is sufficiently large, depending on the location of the points $Y_{s}$, the rate of approximation is estimated by the third Ditzian-Totik modulus of smoothness of $f$ multiplied by a constant $C(s)$, which depends only on $s$.

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  • [BL] R. K. Beatson and D. Leviatan, On comonotone approximation, Canadian Math. Bull. 26 (1983), 220-224. MR 84f:41022
  • [I] G. L. Iliev, Exact estimates for partially monotone approximation, Analysis Math. 4 (1978), 181-197. MR 80h:41007
  • [K] K. A. Kopotun, Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials, Constr. Approx. 10 (1994), 153-178. MR 95k:41014
  • [K1] K. A. Kopotun, Coconvex polynomial approximation of twice differentiable functions, J. Approx. Theory, 83 (1995), 141-156. MR 97e:41043
  • [L] D. Leviatan, Pointwise estimates for convex polynomial approximation, Proc. Amer. Math. Soc. 98 (1986), 471-474. MR 88i:41010
  • [M] Diane Claire Myers, Comonotone and co-convex approximation, Ph.D. dissertation, Temple University: L. Raymon supervisor, 1975.
  • [N] D. J. Newman, Efficient comonotone approximation, J. Approx. Theory 25 (1979), 189-192. MR 80j:41011
  • [NPR] D. J. Newman, E. Passow and L. Raymon, Piecewise monotone polynomial approximation, Trans. Amer. Math. Soc. 172 (1972), 465-472. MR 46:9604
  • [PR] E. Passow and L. Raymon, Monotone and comonotone approximation, Proc. Amer. Math. Soc. 42 (1974), 390-394. MR 89m:41021
  • [PRR] E. Passow, L. Raymon and J. A. Roulier, Comonotone polynomial approximation, J. Approx. Theory 11 (1974), 221-224. MR 50:5293
  • [S] I. A. Shevchuk, Approximation of monotone functions by monotone polynomials, Russ. Akad. Nauk Matem. Sbornik 183 (1992); English transl. in Russ. Acad. Sci. Sbornik Math. 76 (1993), 51-64, MR 94i:41022
  • [S1] I. A. Shevchuk, Approximation by polynomials and traces of functions continuous on an interval, (in Russian), Naukova Dumka, Kiev, Ukraine, 1992.
  • [Sv] A. S. Shvedov, Orders of coapproximation of functions by algebraic polynomials, Mat. Zametki 29 (1981) 117-130; English transl. in Math. Notes 29 (1981), 63-70. MR 82c:41009
  • [Z] S. P. Zhou, On comonotone approximation by polynomials in $L^{p}$ space, Analysis 13 (1993), 363-376. MR 95d:41021

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Additional Information

K. Kopotun
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Address at time of publication: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

D. Leviatan
Affiliation: School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel and Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

I. A. Shevchuk
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv 252601, Ukraine

Keywords: Coconvex polynomial approximation, Jackson estimates
Received by editor(s): May 9, 1996
Received by editor(s) in revised form: April 1, 1997
Additional Notes: The first author acknowledges partial support by the Izaak Walton Killam Memorial Scholarship.
The second author acknowledges partial support by ONR grant N00014-91-1076 and by DoD grant N00014-94-1-1163.
The third author was partially supported by the State Fund for Fundamental Research of Ukraine.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1999 American Mathematical Society

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