Rearrangement invariance and relations among measures of smoothness
Let \(B\) be a Banach space of functions, defined on the \(d\)-dimensional torus or on the unit sphere in \(\mathbb{R} ^d\). Relations between the moduli of smoothness \(\omega ^r(f,t)_B\) and \(\omega ^{r+1}(f,t)_B\) of the sharp Marchaud and sharp lower estimate-type are shown to be satisfied for Banach spaces of functions that are not rearrangement invariant and for which translation or rotation are isometries. Moreover, analogies of the obtained results relating the rate of best approximation with \(\omega ^r(f,t)_B\), that is, the sharp converse inequality and the sharp Jackson inequality are established. For the case that \(B\) is a rearrangement invariant space, these results were obtained earlier by the author and others.
scientific article; zbMATH DE number 6083364
scientific article