International Journal of Mathematics and Mathematical Sciences
Volume 18 (1995), Issue 2, Pages 279-286
On some constants in simultaneous approximation
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 , 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.