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

DOI:
https://doi.org/10.1090/S0002-9939-99-04452-4

MathSciNet review:
1459130

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Abstract | References | Similar Articles | Additional Information

Abstract: Let change its convexity finitely many times in the interval, say times, at . We estimate the degree of approximation of by polynomials of degree , which change convexity exactly at the points . We show that provided is sufficiently large, depending on the location of the points , the rate of approximation is estimated by the third Ditzian-Totik modulus of smoothness of multiplied by a constant , which depends only on .

**[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 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

Email:
kkopotun@math.vanderbilt.edu

**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

Email:
leviatan@math.tau.ac.il

**I. A. Shevchuk**

Affiliation:
Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv 252601, Ukraine

Email:
shevchuk@dad.imath.kiev.ua

DOI:
https://doi.org/10.1090/S0002-9939-99-04452-4

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