  Let $\alpha\in(1,\infty)$ and $\mu$ be a regular finite Borel measure on a
locally compact abelian group. The paper deals with a general trigonometric
approximation problem in $L^\alpha(\mu)$, which arises in prediction theory of
harmonizable symmetric $\alpha$-stable processes. To solve it, a duality method
is applied, which is due to Nakazi and was generalized by Miamee and Pourahmadi
and in the sequel successfully applied by several authors. The novelty of the
present paper is that we do not make any additional assumption on $\mu$.
Moreover, for $\alpha=2$, multivariate extensions are obtained.
