Rational Approximation with Varying Weights, II

✍ Scribed by E.A Rakhmanov; E.B Saff; P.C Simeonov

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 275 KB
• Volume: 92
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1997.3127

✦ Synopsis

We consider two problems concerning uniform approximation by weighted rational functions [w n r n ] n=1 , where r n = p n Âq n has real coefficients, deg p n [:n] and deg q n [;n], for given :>0 and ; 0. For w(x) :=e x we show that on any interval [0, a] with a # (0, a^(:, ;)), every real-valued function f # C([0, a]) is the uniform limit of some sequence [w n r n ]. An implicit formula for a^(:, ;) was given in the first part of this series of papers; in particular, a^(1, 1)=2?. For w(x) :=x % with %>1 we show that uniform approximation of real-valued f # C([b, 1]) on [b, 1] by weighted rationals w n r n is possible for any b # (b (%; :, ;), 1), where b (%; :, ;) was also found in Part I; in particular, b (%; 1, 1)=tan 4 ((?Â4)((%&1)Â%)). Both of the mentioned results are sharp in the sense that approximation is no longer possible if a^is replaced by a^+= or b is replaced by b &= with =>0. We use potential theoretic methods to prove our theorems.