  In this paper we study the sampling recovery problem for certain relevant
multivariate function classes which are not compactly embedded into $L_\infty$.
Recent tools relating the sampling numbers to the Kolmogorov widths in the
uniform norm are therefore not applicable. In a sense, we continue the research
on the small smoothness problem by considering "very" small smoothness in the
context of Besov and Triebel-Lizorkin spaces with dominating mixed regularity.
There is not much known on the recovery of such functions except of an old
result by Oswald in the univariate situation. As a first step we prove the
uniform boundedness of the $\ell_p$-norm of the Faber-Schauder coefficients in
a fixed level. Using this we are able to control the error made by a (Smolyak)
truncated Faber-Schauder series in $L_q$ with $q<\infty$. It turns out that the
main rate of convergence is sharp. As a consequence we obtain results also for
$S^1_{1,\infty}F([0,1]^d)$, a space which is ``close'' to the space
$S^1_1W([0,1]^d)$ which is important in numerical analysis, especially
numerical integration, but has rather bad Fourier analytic properties.
