Uniform rational approximation with osculatory interpolation

✍ Scribed by A.L. Perrie

• Publisher: Elsevier Science
• Year: 1970
• Tongue: English
• Weight: 673 KB
• Volume: 4
• Category: Article
• ISSN: 0022-0000
• DOI: 10.1016/s0022-0000(70)80026-5

✦ Synopsis

The purpose of this paper is to extend several results in the theory of generalized rational approximation. Let X be a compact space and for f ~ C(X) define Ilfll ~ max{Jf(x)l : x E X}.

Suppose that P and Q are two finite-dimensional subspaces of C(X). Then, in generalized rational approximation, one is interested in approximating an fe C(X) by a function of the form r = p/q wherep ~ P, q e Q, and q > 0 on X.

The problem that we wish to consider in this paper is that of approximating a functionf by a k-point osculating rational function. To be specific, let X be a closed subset of the closed interval [a, b], {Ya,.--, Yk} a fixed set of k points in X, {m a ..... mk) k a set of positive integers with m* = ~2i=1 mi, and s = maxi {m, --1}. Suppose P and Q are two finite-dimensional subspaces of Cs(X). We define the set R to be R = {r =p/q :pEP, q~Q, q > 0 on x}, and forf~ C*(X) we define the set R(f) to be

Then we are interested in finding a best approximation to an f~ C~(X) from the set R(f). Forf~ Cs(X) we define Ilfll ----max{If(x)l : x ~ X} and we call r* ~ R(f) a best approximation toffrom R(f) iff