On some constants in simultaneous approximation

Abstract

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.