Moduli of Smoothness and Rate of A.e. Convergence for Some Convolution Operators

One purpose of this article is to establish results on the rate of almost everywhere convergence of approximation processes of convolution type in L(R), where instead of a particular rate (like t, μ > 0, t → 0+) fractional moduli of smoothness are employed. An essential tool is a modified K-functional. Away from saturation orders these results are nearly optimal. A second purpose is to illustrate that the methods applied also work in other settings which feature a convolution/multiplier structure.