ON SOME CONSTANTS IN SIMULTANEOUS APPROXIMATION

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.


INTRODUCTION
There are several results which give pointwise estimates of the error in simultaneous approximation of a function f E Cq[-1, 1] and of its q derivatives by an algebraic polynomial and its corresponding derivatives, in terms of the modulus of continuity of f(q), which we recall is defined by w(f('); )= suPl,-,l<lf(')(:r,) One of the first of these results was a theorem of Trigub, the relevant part of which states: THEOREM 1. (see Trigub [I]) Let f E C'[ -1, 1] Then .for each n > 2q, there exists a poly,tom,al P, of degree at most n such that for k O,...,q and for -1 <_ z <_ + f(q).v/1 x'-' with M independent of n and f. ,) + (.)A theorem of Gopengauz [2], based upon the theorem of Trigub, shows that there is the possibility of exact approximation at tim endpoints :t:l" THEOREM 2. Let f E Cq[ -1, 1].Then for each n > 4q + 5, there eztsts a polynom,al at most n such that for k O ,q and for-1 < x < (1.) with It" independent of n and f.Such results as these were originally pursued for the sake of completing the theory of algebraic polyno- mial al)proximation.Theoretical interest in similar results has been widespread and sustained, leadig to a literature too extensive to cite or l)araphrase here.In such a theoretical context, however, the question of obtaining a value for the constants is not urgent.In fact a close examination of the original proofs of these results and of those in such related work as that of Telyakovskii [3] shows that the proofs are ex- tremely uneconomical concerning constants, and the question of estimating auy of tle relevant constants (obviously difficult) has been little addressed in the subsequent work.
More recently, results such as the theorem of Gopengauz have been used as essential tools in showing that Lagrange interpolation (and other linear projections, too) can be modified by interpolation of some derivatives at +1 with good effect: the derivatives of the function being approximated are simultaneously approximated by the derivatives of the approximating polynomial, at a rate which approaches what is theoretically feasible.Evidence obtained from computer experimentation indicates that polynomial inter- polation e.g. on nodes generated by Chebyshev or Jacobi polynomials, when ,no(lifted by the introduction of interpolation of derivatives at +1, can give numerical results for convergence to many standard "bad" functions (the infamous Runge function and some others) and their derivatives which a're very good indeed.For example, Tasche [4] an(l llaszenski and Tasche [5] have combined these methods with cotn- putation by a fast algorithm and have developed a method of approximation which seems comparable in its practical efficiency to cubic spline approximation and seems to give superior approximation for a comparable number of data points if the function is several times differentiable. Thus, there is the potential of applying new methods of simultaneous approximation in nunerical mathematics, but estimates of the error incurred in approximation require reasonable estinates of the constants in such basic theorems on simultaneous approximation as that of Gopcngauz.l[ere, we begin to address this problem with the following result: THEOREM 3.For a function f C(q)[-1, 1] let P,, be a poly,,o,nial satisfy,ng (1.1) for some C (which may or may not depend upon n or f, as we choose) and also satisfying f()(-l-1) P,*)(-t-1) for k 0,...,q. (1.3) Then (1.2) follows with a constant h" <_ max(,le'C, 7C + 7).In particvlar, the relation between h" and C is absolute and independent of all other quantities involved.
From our theorem, the theorem of Gopengauz will also follow, once it is shown how to construct poly- nomials satisfying (1.1) and (1.3).We will complete the proof of (1.2) by giving a new construction for such polynomials.Thus, (1.2) will be derived directly from Trigub's (1.1), bypassing a second conclusion of Trigub's theorem (not stated here in detail) giving a pointwise estimate for I;?+')(z)l.There are two reasons for this innovation.One is that the second conclusion of Trigub, while interesting in itself, is not needed here.The second is that for both the first and the second conclusion of Trigub's theorem to hold, the value of the constant M in the theorem must be large enough to accommodate both conclusions, requiring the user of Trigub's theorem to begin his work with a larger value of M. Another fact relevant here is that the original proof of Gopengauz and also the derivation of a pointwise estimate for p,?+l in Trigub's theorem depend upon an inequality of Brudnyi giving such pointwise estimates of derivatives.
To date, no estimate of the constant in Brudnyi's inequality exists, either.Our proof will permit another improvement which can have importance in applications: the minimal value of n for which the estimate is valid can be lowered from 4q + 5 to 2q + 1.
For z 0 we thus obtain Finally, since , was arbitrary, we may set ,(z) T,,(z + 0), obtaining (2.1), and the proof of the Lemma is complete, l-I

PROOFS
We begin with the proof of Theorem 3, and then we will show how this result can be used in the proof of Theorem 2.
PROOF: The proof of our Theorem (Theorem 3) divides itself naturally, into two cases: that k q (including the possibility that k q 0 is one of the cases, and that k < q is the other.Case 1 (k q): We prove that If')(+l) P.'(+I)I 0 and If'(x)-P.c'(x)I< C w f('); +implies vzi'-:) in which C depends only upon C. The special case that q 0 is also covered here.
This argument completes the proof of Case 2.
The constant K in Theorem can be defined as max{C',C"}, and Theorem a is proven.