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

## TL;DR

This paper explores spectral analysis methods within specific functional spaces to solve inhomogeneous linear functional equations, extending techniques traditionally used for homogeneous equations and applying them to problems motivated by numerical integration.

## Contribution

It introduces a novel application of spectral analysis to inhomogeneous functional equations in translation invariant subspaces, expanding the theoretical framework beyond homogeneous cases.

## Key findings

- Spectral analysis effectively solves certain inhomogeneous linear functional equations.
- The approach extends existing methods from homogeneous to inhomogeneous equations.
- Applications include problems motivated by quadrature rules in numerical integration.

## Abstract

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.