International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 2, Pages 279-286

On some constants in simultaneous approximation

K. Balázs1 and T. Kilgore2

1Budapest University of Economics, Pf. 489, Budapest 5 H-1828, Hungary
2Division of Mathematics, Auburn University, Auburn 36849, Alabama, USA

Received 3 March 1993; Revised 18 February 1994

Copyright © 1995 K. Balázs and T. Kilgore. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Pointwise estimates for the error which is feasible in simultaneous approximation of a function and its derivatives by an algebraic polynomial were originally pursued from theoretical motivations, which did not immediately require the estimation of the constants in such results. However, recent numerical experimentation with traditional techniques of approximation such as Lagrange interpolation, slightly modified by additional interpolation of derivatives at ±1, shows that rapid convergence of an approximating polynomial to a function and of some derivatives to the derivatives of the function is often easy to achieve. The new techniques are theoretically based upon older results about feasibility, contained in work of Trigub, Gopengauz. Telyakovskii, and others, giving new relevance to the investigation of constants in these older results. We begin this investigation here. Helpful in obtaining estimates for some of the constants is a new identity for the derivative of a trigonometric polynomial, based on a well known identity of M. Riesz. One of our results is a new proof of a theorem of Gopengauz which reduces the problem of estimating the constant there to the question of estimating the constant in a simpler theorem of Trigub used in the proof.