Che by she v Approximations of a Function

Che by she v Approximations of a Function and Its Derivatives By D. G. Moursund 1. Introduction. This paper considers the problem of the of a function and its first r derivatives. Several theorems and nature of the extremals of a best approximation are are applied to a special case of approximating a function and a uniqueness theorem is obtained.

uniform approximation concerning the number obtained. These results and its first derivative,

2. Statement of the Problem. Let X be a compact subset of the real line. Let n S; 0 and r Sï 1 be fixed integers. The function /(x), which is to be approximated, and the base functions oix),iix), ■■■ ,i, ■• ■ , 0 there exist points xx, x2 in X such that x0 — e < xx < x0 < x2 < Xo + e then DLk(x, a) = 0 at the point x0. Proof. Suppose that Lk(x0, a) = d. Then nr / \ r Lk(x, a) — d DLk(Xo, a) = Inn. x-*Xq

x

a;o

For all x, Lk(x, a) — d ^ 0. Hence if x < x0 then (7*;(a;, a) — d)/(x — x0) ^ 0 while if x > xo then (Lk(x, a) — d)/(x — xQ) i£ 0. Since the approach to the limit

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APPROXIMATION OF A FUNCTION AND ITS DERIVATIVES

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may be made from either side of x0, through points of X, it follows that the limit is zero. A similar argument holds if Lkixo, a) = —d so the proof is complete. 22. Corollary. Suppose that Pix, a) is a best approximation to f, with a in the interior of R, and that the hypotheses of Theorem 21 are satisfied at the point (xa, k). Then for any other best approximation P(x, ß) it follows that Dk+1P(x0,a)

= Dk+1P(xo,ß).

Proof. From Corollary 18 we know that (x0 ,k) is an extremal of the approximation P(a;, ß) to/. Hence DLk(x0,a)

= 0 = DLk(x0,ß).

Using the product rule for differentiation

we have

Dwk(xo)Dk[P(xo,

a) - f(xo)] + wk(xo)Dk+1[P(xo,a)

Dwk(xo)Dk[P(xo,

ß) - f(xo)] + Wk(xo)Dk+1[P(xo, ß) - f(xü)] = 0.

Since Dk[Pix0,a)

- f(x0)] = 0,

- fix*)] = Dk[Pix0, ß) - fix0)] and wkix0) ^ 0 the result is

established. 5. Uniqueness. The previous theorems give us considerable information about the extremals of a best approximation. We shall use these results to establish the

following theorem. Let X = [—1, 1], r = l,