Some Positive Results and Counterexamples in Comonotone Approximation

✍ Scribed by D. Leviatan; I.A. Shevchuk

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 320 KB
• Volume: 89
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1997.3038

✦ Synopsis

Let f be a continuous function on [&1, 1], which changes its monotonicity finitely many times in the interval, say s times. We discuss the validity of Jackson-type estimates for the approximation of f by algebraic polynomials that are comonotone with it. While we prove the validity of the Jackson-type estimate involving the Ditzian Totik modulus of continuity and a constant which depends only on s, we show by counterexamples that in many cases this is not so, even for functions which possess locally absolutely continuous derivatives. These counterexamples are given when there are certain relations between s, the number of changes of monotonicity, and r, the number of derivatives. For other cases we do have some Jackson-type estimates and another paper will be devoted to that.

1997 Academic Press where E (1) n ( f ) denotes the degree of approximation of f by nondecreasing algebraic polynomials of degrees n, c an absolute constant and |( f, t) the modulus of continuity of f. article no. AT963038 195