  The aim of the paper is to investigate the solutions of special inhomogeneous
linear functional equations by using spectral analysis in a translation
invariant closed linear subspace of additive/multiadditive functions containing
the restrictions of the solutions to finitely generated fields. The application
of spectral analysis in some related varieties is a new and important trend in
the theory of functional equations; especially they have successful
applications in case of homogeneous linear functional equations. The foundation
of the theory can be found in M. Laczkovich and G. Kiss \cite{KL}, see also G.
Kiss and A. Varga \cite{KV}. We are going to adopt the main theoretical tools
to solve some inhomogeneous problems due to T. Szostok \cite{KKSZ08}, see also
\cite{KKSZ} and \cite{KKSZW}. They are motivated by quadrature rules of
approximate integration.
