Equivalence of rate of approximation and smoothness (Q804854)

The result of Jackson and Bernstein \[ \omega^ r(f,t)=O(t^{\alpha})\Leftrightarrow E_ n(f)=O(n^{-\alpha}) \] have led to the question whether \(\omega^ r(f,\frac{1}{n})\sim E_ n(f)?\) The answer is affirmative, if the function \(\psi_ r(t):=\omega^ r(f,t)\) satisfies \[ \delta^ r\int^{c}\frac{\psi_ r(t)}{t^{r+1}}dt\sim \psi_ r(\delta). \] The analogous result for the Peetre functional is also known. In this paper analogous results for some nonlinear processes of best approximation are achieved. Moreover, weighted algebraic polynomial approximation is investigated and moduli that do not satisfy the Jackson inequality but a weaker inequality are considered.