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 [2], see also G. Kiss and A. Varga [1]. We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to T. Szostok [5], see also [6] and [7]. They are motivated by quadrature rules of approximate integration.

Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and automorphisms are additive functions satisfying some further functional equations as well. It is an important question that how these morphisms can be characterized among additive mappings in general.The paper contains some multivariate characterizations of higher order derivations. The univariate characterizations are given as consequences by the diagonalization of the multivariate formulas. This method allows us to refine the process of computing the solutions of univariate functional equations of the form n k=1xwhere p k and q k (k = 1, . . . , n) are given nonnegative integers and the unknown functions f 1 , . . . , f n : R → R are supposed to be additive on the ring R. It is illustrated by some explicit examples too.As another application of the multivariate setting we use spectral analysis and spectral synthesis in the space of the additive solutions to prove that it is spanned by differential operators. The results are uniformly based on the investigation of the multivariate version of the functional equations.

Abstract. For a wide class of linear functional equations the solutions are generalized polynomials. The existence of non-trivial monomial terms of the solution strongly depends on the algebraic properties of some related families of parameters. As a continuation of the previous work [A. Varga, Cs. Vincze, G. Kiss, Algebraic methods for the solution of linear functional equations, Acta Math. Hungar.] we are going to present constructive algebraic methods of the solution in some special cases. Explicit examples will be also given.

As a continuation of our previous work [18] 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 [3]. 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 C and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration [6], see also [7] and [8]. Introduction and preliminariesLet C denote the field of complex numbers. We are going to investigate the family of functional equations of typewhere α i and β i (i = 1, . . . , n) are given real or complex parameters, p = 1, . . . , 2n − 1 and c p ∈ C is a constant depending on p. 1 reasons we pay a special attention to the case of p = 1, i.e.where c 1 is rewritten as c for the sake of simplicity. The problem of solving the family of equations (1) as p runs through its possible values 1, . . . , 2n − 1 is equivalent to the solution of equationwhere x, y ∈ C and f, F : C → C are unknown functions. It is motivated by quadrature rules of approximate integration [6], see also [7] and [8].Remark 1.1. In order to substitute x = 0 or y = 0 into (1) we agree that 0 0 := 1. Such a special choice reproduces the pair of equationsthat are consequences of (3) for monomial solutions of degree p. For a more detailed survey of the preliminary results see [18].Let (G, * ) be an Abelian group; C G denotes the set of complex valued functions defined on G. A function f : G → C is a generalized polynomial, if there is a non-negative integer p such thatfor any g 1 , . . . , g p+1 ∈ G. Here ∆ g is the difference operator defined by, where f ∈ C G and g ∈ G. The smallest p for which (5) holds for any g 1 , . . . , g p+1 ∈ G is the degree of the generalized polynomial f . A function F :for any x ∈ G. It is known that any generalized polynomial function can be written as the sum of generalized monomials [11]. By a general result of M. Sablik [10] any solution of (3) is a generalized polynomial of degree at most 2n − 1 under some mild conditions for the parameters in the functional equation:see also Lemma 2 in [7]. The generalized polynomial solutions of (3) are constituted by the sum of the diagonalizations of p-additive functions satisfying equations of type (1).In what follows we are going to use spectral synthesis in translation invariant closed linear subspaces of additive functions on some finitely generated 2 fields containing the restrictions of the solutions of functional equati...

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