Approximation by weighted polynomials

✍ Scribed by David Benko

• Book ID: 104142787
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 262 KB
• Volume: 120
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1016/s0021-9045(02)00017-5

✦ Synopsis

It is proven that if xQ 0 ðxÞ is increasing on ð0; þNÞ and wðxÞ ¼ expðÀQðxÞÞ is the corresponding weight on ½0; þNÞ; then every continuous function that vanishes outside the support of the extremal measure associated with w can be uniformly approximated by weighted polynomials of the form w n P n : This problem was raised by Totik, who proved a similar result (the Borwein-Saff conjecture) for convex Q: A general criterion is introduced, too, which guarantees that the support of the extremal measure is an interval. With this criterion we generalize the above approximation theorem as well as that one, where Q is supposed to be convex.