Inverse Rational L1 Approximation

✍ Scribed by R.C. Gayle

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 247 KB
• Volume: 83
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1995.1109

✦ Synopsis

We consider the nonlinear approximating family (R_{m}^{n}) of rational expressions over a real interval. In the (L^{p}) norms, (l<p<\infty) non-normal elements of this family cannot arise as best approximations to functions outside the family. In the (L^{\prime}) case, Dunham (1971) has shown that for a continuous function no rational of defect two or greater, excepting the rather special case of the function 0 , can be a best approximation. Cheney and Goldstein have shown (1967) that any normal rational function can arise as the best approximation to some function (f \in L^{2}) which is not in the rational family. We show here that there exist continuous functions not in (R_{m}^{n}) which do have any given defect one functions as their best approximations by using variational techniques from Wolle (1976). " 1995 Academic Press. Inc.