# On spectral synthesis in varieties containing the solutions of inhomogeneous linear functional equations

## TL;DR

This paper extends spectral synthesis techniques to characterize solutions of inhomogeneous linear functional equations within certain algebraic structures, revealing they are spanned by exponential monomials linked to automorphisms and differential operators.

## Contribution

It introduces a spectral synthesis approach to fully describe solutions of specific inhomogeneous linear functional equations in finitely generated fields, building on prior spectral analysis methods.

## Key findings

- Solutions are spanned by exponential monomials.
- Spectral analysis confirms existence of special solutions.
- Method applies to problems in approximate integration.

## Abstract

As a continuation of our previous work \cite{KV2} the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation invariant closed linear subspaces of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The idea is based on the fundamental work of M. Laczkovich and G. Kiss \cite{KL}. Using spectral analysis in some related varieties we can prove the existence of special solutions (automorphisms) of the functional equation but the spectral synthesis allows us to describe the entire space of solutions on a large class of finitely generated fields. It is spanned by the so-called exponential monomials which can be given in terms of automorphisms of $\cc$ and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration \cite{KKSZ08}, see also \cite{KKSZ} and \cite{KKSZW}.