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Published online by Cambridge University Press: 20 November 2018
Averages in weighted spaces
$L_{\phi }^{p}[-1,1]$
defined by additions on
$[-1,\,1]$ will be shown to satisfy strong converse inequalities of type
$\text{A}$ and
$\text{B}$ with appropriate
$K$-functionals. Results for higher levels of smoothness are achieved by combinations of averages. This yields, in particular, strong converse inequalities of type
$\text{D}$ between
$K$-functionals and suitable difference operators.