The degree of coconvex polynomial approximation

Kopotun, K.; Leviatan, D.; Shevchuk, I.

doi:10.1090/S0002-9939-99-04452-4

The degree of coconvex polynomial approximation
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by K. Kopotun, D. Leviatan and I. A. Shevchuk PDF

Proc. Amer. Math. Soc. 127 (1999), 409-415 Request permission

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 $s$.

References

R. K. Beatson and D. Leviatan, On comonotone approximation, Canad. Math. Bull. 26 (1983), no. 2, 220–224. MR 697804, DOI 10.4153/CMB-1983-034-0
G. L. Iliev, Exact estimates for partially monotone approximation, Anal. Math. 4 (1978), no. 3, 181–197 (English, with Russian summary). MR 514758, DOI 10.1007/BF01908988
Kirill A. Kopotun, Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials, Constr. Approx. 10 (1994), no. 2, 153–178. MR 1305916, DOI 10.1007/BF01263061
Kirill A. Kopotun, Coconvex polynomial approximation of twice differentiable functions, J. Approx. Theory 83 (1995), no. 2, 141–156. MR 1357583, DOI 10.1006/jath.1995.1113
D. Leviatan, Pointwise estimates for convex polynomial approximation, Proc. Amer. Math. Soc. 98 (1986), no. 3, 471–474. MR 857944, DOI 10.1090/S0002-9939-1986-0857944-9
Diane Claire Myers, Comonotone and co-convex approximation, Ph.D. dissertation, Temple University: L. Raymon supervisor, 1975.
D. J. Newman, Efficient co-monotone approximation, J. Approx. Theory 25 (1979), no. 3, 189–192. MR 531408, DOI 10.1016/0021-9045(79)90009-1
D. J. Newman, Eli Passow, and Louis Raymon, Piecewise monotone polynomial approximation, Trans. Amer. Math. Soc. 172 (1972), 465–472. MR 310506, DOI 10.1090/S0002-9947-1972-0310506-9
Rui Zhi Liu, Comment: “Monotone and comonotone approximation” [Proc. Amer. Math. Soc. 42 (1974), 390–394; MR0336176 (49 #952)] by E. Passow and L. Raymon, J. Math. Res. Exposition 9 (1989), no. 1, 145–148 (Chinese, with English summary). MR 984742
Eli Passow, Louis Raymon, and John A. Roulier, Comonotone polynomial approximation, J. Approximation Theory 11 (1974), 221–224. MR 352807, DOI 10.1016/0021-9045(74)90013-6
I. A. Shevchuk, Approximation of monotone functions by monotone polynomials, Mat. Sb. 183 (1992), no. 5, 63–78 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 76 (1993), no. 1, 51–64. MR 1184310, DOI 10.1070/SM1993v076n01ABEH003401
I. A. Shevchuk, Approximation by polynomials and traces of functions continuous on an interval, (in Russian), Naukova Dumka, Kiev, Ukraine, 1992.
A. S. Švedov, Orders of coapproximations of functions by algebraic polynomials, Mat. Zametki 29 (1981), no. 1, 117–130, 156 (Russian). MR 604156
S. P. Zhou, On comonotone approximation by polynomials in $L^p$ space, Analysis 13 (1993), no. 4, 363–376. MR 1256520, DOI 10.1524/anly.1993.13.4.363