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On Positive and Copositive Polynomial and Spline Approximation inLp[−1, 1], 0<p<∞

✍ Scribed by Y.K. Hu; K.A. Kopotun; X.M. Yu

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 610 KB
Volume: 86
Category: Article
ISSN: 0021-9045
DOI: 10.1006/jath.1996.0073

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✦ Synopsis

For a function f # L p [&1, 1], 0< p< , with finitely many sign changes, we construct a sequence of polynomials P n # 6 n which are copositive with f and such that & f &P n & p C| . ( f , (n+1) &1 ) p , where | . ( f , t) p denotes the Ditzian Totik modulus of continuity in L p metric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, then | . cannot be replaced by | 2 if 1< p< . In fact, we show that even for positive approximation and all 0< p< the same conclusion is true. Also, some results for (co)positive spline approximation, exact in the same sense, are obtained.

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