  We study approximation properties of linear sampling operators in the spaces
$L_p$ for $1\le p<\infty$. By means of the Steklov averages, we introduce a new
measure of smoothness that simultaneously contains information on the
smoothness of a function in $L_p$ and discrete information on the behaviour of
a function at sampling points. The new measure of smoothness enables us to
improve and extend several classical results of approximation theory to the
case of linear sampling operators. In particular, we obtain matching direct and
inverse approximation inequalities for sampling operators in $L_p$, find the
exact order of decay of the corresponding $L_p$-errors for particular classes
of functions, and introduce a special $K$-functional and its realization
suitable for studying smoothness properties of sampling operators.

Classical Analysis and ODEs;Numerical Analysis;Numerical Analysis