Boundary properties of functions harmonic in a strip

We study in the $L_p$-norm, $1\le p\le\infty$, the boundary properties of the solution to the Dirichlet problem for the strip
$$ \mathscr A=\{(x,y):-\infty<x<\infty,\ 0<y<\eta,\ \eta>0\} $$
and its dependence on the structural properties of the given boundary values (symmetric, antisymmetric). In particular, for the case of symmetric boundary values we obtain direct and inverse theorems on approximation in terms of the general modulus of continuity of second order.