  Let $\Gamma$ be an LCA group and $(\mu_n)$ be a sequence of bounded regular
Borel measures on $\Gamma$ tending to a measure $\mu_0$. Let $G$ be the dual
group of $\Gamma$, $S$ be a non-empty subset of $G \setminus \{ 0 \}$, and
$[{\mathcal T}(S)]_{\mu_n,p}$ the subspace of $L^p(\mu_n)$, $p \in (0,\infty)$,
spanned by the characters of $\Gamma$ which are generated by the elements of
$S$. The limit behaviour of the sequence of metric projections of the function
$1$ onto $[{\mathcal T}(S)]_{\mu_n,p}$ as well as of the sequence of the
corresponding approximation errors are studied. The results are applied to
obtain stability theorems for prediction of weakly stationary or harmonizable
symmetric $p$-stable stochastic processes. Along with the general problem the
particular cases of linear interpolation or extrapolation as well as of a
finite or periodic observation set are studied in detail and compared to each
other.
