Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials |
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Authors: | Kirill A Kopotun |
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Institution: | 1. Department of Mathematics, University of Alberta, T6G 2G1, Edmonton, Alberta, Canada
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Abstract: | Let Δ q be the set of functionsf for which theqth difference, is nonnegative on the interval ? 1,1],P n is the set of algebraic polynomials of degree not exceedingn, τ k (f, δ) p is the averaged Sendov-Popov modulus of smoothness in theL p ?1,1] metric for 1≦p≦∞, ω k (f, δ) and $\omega _\phi ^k (f,\delta ),\phi (x): = \sqrt {1 - x^2 } ,$ , are the usual modulus and the Ditzian-Totik modulus of smoothness in the uniform metric, respectively. For a functionf∈C?1,1]?Δ2 we construct a polynomialp n ∈P n ?Δ2 such that $$\begin{gathered} \left| {f(x) - p_n (x)} \right| \leqslant C\omega _3 (f,n^{ - 1} \sqrt {1 - x^2 } + n^{ - 2} ),x \in - 1,1]; \hfill \\ \left\| {f - p_n } \right\|_\infty \leqslant C\omega _\phi ^3 (f,n^{ - 1} ); \hfill \\ \left\| {f - p_n } \right\|_p \leqslant C\tau _3 (f,n^{ - 1} )_p . \hfill \\ \end{gathered}$$ As a consequence, for a functionf∈C 2?1,1]?Δ3 a polynomialp n * ∈P n ?Δ3 exists such that $$\left\| {f - p_n^* } \right\|_\infty \leqslant Cn^{ - 1} \omega _2 (f\prime ,n^{ - 1} ),$$ wheren≥2 andC is an absolute constant. |
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