Pointwise estimates for monotone polynomial approximation
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We prove that iff is increasing on [−1,1], then for eachn=1,2,... there is an increasing algebraic polynomialP n of degreen such that |f(x)−P n (x)|≤cω2(f,√1−x2/n), whereω2 is the second-order modulus of smoothness. These results complement the classical pointwise estimates of the same type for unconstrained polynomial approximation. Using these results, we characterize the monotone functions in the generalized Lipschitz spaces through their approximation properties.
1980 AMS(MOS) subject classifications41 A 10 41 A 25 41 A 29
Key words and phrasesMonotone approximation by algebraic polynomials Degree of approximation Pointwise estimates
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