Positive and co-positive Lp approximation with interpolation constraint

Dreeb, Nabiha Kahtan; Bhaya, Eman Samir

doi:10.47974/JIM-2311

Open Access Research Article

Positive and co-positive Lp approximation with interpolation constraint

^*Nabiha Kahtan DreebCorresponding authornabihakahtan@gmail.comDepartment of Mathematics College of Education for Pure Sciences University of BabylonBabylon, 51002, IraqView full profile → , Eman Samir Bhayaemanbhaya@alzahraa.edu.iqDepartment of Mathematics College of Education Al-Zahraa University for WomenDepartment of Mathematics College of Education Al-Zahraa University for WomenKarbala, IraqView full profile →

* Corresponding author · click or hover a name for details

pp. 1279–1290Vol. 29Issue 5-AMay 2026DOI: 10.47974/JIM-2311XML

Received:: 01 May 2025
Published Online:: 27 Jun 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2311
Pages:: 1279–1290

Abstract

It is well known, that if f be continuous functions define on [a, b], a, b ∈ R, there is interpolating polynomials Pn on {ai}m i= 1 satisfy: ‖f – Pn ‖p ≤ 1/nr ωt (f(r),1/n)p, where ωt is the t – th usual modulus for the smoothness’ , and Pn(i) (ai) = f(i) (ai), 1 ≤ i ≤ m, m < r, m, r ∈ N. In this paper we shall answer the following questions: Is the above result true for intertwining, one-sided, positive, co-positive approximation.

Keywords

Primary 93A30 Secondary 49K15

Subject Classifications

Positive approximationCo-positive approximationOne-sided approximationIntertwining approximation

References

[1] K. A. Kopotun, “On k-monotone polynomial and spline approximation in Lp, 0 < p < ∞(quasi-norm),” in Approximation Theory VIII: Approximation and Interpolation, C. K. Chui and L. L. Schumaker, Eds., vol. 1. Singapore: World Scientific, pp. 295–302 (1995).
[2] K. A. Kopotun, D. Leviatan, A. Prymak, and I. A. Shevchuk, “Uniform and pointwise shape preserving approximation by algebraic polynomials,” Surveys in Approximation Theory, vol. 6, pp. 24–74 (2011).
[3] H. J. Kushner and D. S. Clark, Stochastic Approximation Methods for Constrained and Unconstrained Systems, vol. 26. Springer Science & Business Media (2012).
[4] Y. K. Hu, K. A. Kopotun, and X. M. Yu, “Constrained approximation in Sobolev spaces,” Canadian Journal of Mathematics, vol. 49, no. 1, pp. 74–99 (1997).
[5] P. Bullen, “A criterion for n-convexity,” Pac. J. Math., vol. 36, no. 1, pp. 81–98 (1971).
[6] S. P. Zhou, “A counterexample in copositive approximation,” Israel Journal of Mathematics, vol. 78, no. 1, pp. 75–83 (1992).
[7] S. P. Zhou, “On copositive approximation,” Approximation Theory and its Applications, vol. 9, no. 2, pp. 104–110 (1993).
[8] M. J. D. Powell, Approximation Theory and Methods. Cambridge, UK: Cambridge University Press, pp. 352 (1981).
[9] K. A. Kopotun, D. Leviatan, and I. A. Shevchuk, “Exact order of pointwise estimates for polynomial approximation with Hermite interpolation,” J. Approx. Theory, vol. 264, p. 105538 (2021).
[10] V. K. Dzyadyk and I. A. Shevchuk, Theory of Uniform Approximation of Functions by Polynomials. Berlin & New York, USA: Walter de Gruyter, pp. 480 (2008).
[11] G. A. Dzyubenko, “Copositive pointwise approximation,” Ukrainian Mathematical Journal, vol. 48, no. 3, pp. 367–376 (1996).
[12] G. A. Dzyubenko, “Interpolated estimate for copositive approximations by algebraic polynomials,” Ukrainian Mathematical Journal, vol. 74, no. 4, pp. 563–574 (2022).
[13] S. A. Al-Ameedee and A. J. Obaid, “New results and application of differential quasi subordinations for higher-order derivatives of meromorphic multivalent functions,” Journal of Interdisciplinary Mathematics, vol. 26, no. 4, pp. 643–650 (2023), doi: 10.47974/JIM-1483.
[14] S. A. Al-Ameedee and A. J. Obaid, “Sandwich results of meromorphic multivalent functions defined by multiplier transform,” Journal of Interdisciplinary Mathematics, vol. 26, no. 4, pp. 651–658 (2023), doi: 10.47974/JIM-1484.

Views: 52Downloads: 6Citations: 0