Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials

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.