Hoffman's Error Bounds and Uniform Lipschitz Continuity of Bestlp-Approximations

✍ Scribed by H Berens; M Finzel; W Li; Y Xu

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 272 KB
• Volume: 213
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.1997.5521

✦ Synopsis

In a central paper on smoothness of best approximation in 1968 R. Holmes and B. Kripke proved among others that on ‫ޒ‬ n , endowed with the l -norm, 1p -ϱ, p the metric projection onto a given linear subspace is Lipschitz continuous where the Lipschitz constant depended on the parameter p. Using Hoffman's Error Bounds as a principal tool we prove uniform Lipschitz continuity of best l -app proximations. As a consequence, we reprove and prove, respectively, Lipschitz Ž . continuity of the strict best approximation sba, p s ϱ and of the natural best Ž . approximation nba, p s 1 .