# Copositive pointwise approximation

Article

Received:

- 23 Downloads
- 1 Citations

## Abstract

We prove that if a function, where ω

*f*∈*C*^{(1)}(*I*),*I*: = [−1, 1], changes its sign*s*times (*s*∈ ℕ) within the interval*I*, then, for every*n*>*C*, where*C*is a constant which depends only on the set of points at which the function changes its sign, and*k*∈ ℕ, there exists an algebraic polynomial*P*_{ n }=*P*_{ n }(*x*) of degree ≤*n*which locally inherits the sign of*f(x)*and satisfies the inequality$$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$

_{ k }(*f*′;*t*) is the*k*th modulus of continuity of the function*f*’. It is also shown that if*f*∈*C*(*I*) and*f*(*x*) ≥ 0,*x*∈*I*then, for any*n*≥*k*− 1, there exists a polynomial*P*_{ n }=*P*_{ n }(*x*) of degree ≤*n*such that*P*_{ n }(*x*) ≥ 0,*x*∈*I*, and |*f*(*x*) −*P*_{ n }(*x*)| ≤*c*(*k*)ω_{ k }(*f*;*n*^{−2}+*n*^{−1}√1 −*x*^{2}),*x*∈*I*.## Keywords

Polynomial Kernel Pointwise Approximation Algebraic Polynomial Spline Approximation Nonincreasing Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.E. Passow and L. Raymon, “Copositive polynomial approximation,”
*J. Approx. Theory*,**12**, 299–304 (1974).MATHCrossRefMathSciNetGoogle Scholar - 2.J. A. Roulier, “The degree of copositive approximation,”
*J. Approx. Theory*,**19**, 253–258 (1977).MATHCrossRefMathSciNetGoogle Scholar - 3.D. Leviatan, “The degree of copositive approximation by polynomials,”
*Proc. Amer. Math. Soc.*,**88**, 101–105 (1983).MATHCrossRefMathSciNetGoogle Scholar - 4.Y. K. Hu, D. Leviatan, and X. M. Yu, “Copositive polynomial approximation in C [-1, 1],”
*J. Analysis*,**1**, 85–90 (1993).MATHMathSciNetGoogle Scholar - 5.K. A. Kopotun, “On copositive approximation by algebraic polynomials, ”
*Anal. Math*, (to appear).Google Scholar - 6.Y. K. Hu and X. M. Yu, “The degree and algorithm of copositive approximation,”
*SIAM Anal.*(to appear).Google Scholar - 7.S. P. Zhou, “On copositive approximation,”
*SIAM Anal.*(to appear).Google Scholar - 8.S. P. Zhou, “A counter example in copositive approximation,”
*Israel J. Math.*,**78**, 75–83 (1992).MATHCrossRefMathSciNetGoogle Scholar - 9.X. M. Yu, “Degree of copositive polynomial approximation,”
*Chin. Ann. Math.*,**10**, 409–415 (1989).MATHGoogle Scholar - 10.Y. K. Hu, D. Leviatan, and X. M. Yu, “Copositive polynomial and spline approximation,”
*J. Approx. Theory*,**80**, 204–218 (1995).MATHCrossRefMathSciNetGoogle Scholar - 11.A. G. Dzyubenko, J. Gilewicz, and I. A. Shevchuk,
*Piecewise Monotone Pointwise Approximation*, Preprint CPT-94/P. 3121, CNRS Lumini, Marseilles (1994).Google Scholar - 12.I. A. Shevchuk,
*Polynomial Approximation and Traces of Functions Continuous on an Interval*[in Russian], Naukova Durnka, Kiev 1992.Google Scholar - 13.V. K. Dzyadyk, “On constructive description of the functions satisfying the condition [Lip α, (0 < α < 1)] on a finite segment of the real axis,”
*Izv. Akad. Nauk SSSR, Ser. Mat.*,**20**, No. 2, 623–642 (1956).MathSciNetGoogle Scholar

## Copyright information

© Plenum Publishing Corporation 1997