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

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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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In this paper, we consider the condition $$\sum _{i=0}^{n+1}\varphi _i(r_ix+q_iy)\in {\mathbb {Z}}$$ ∑ i = 0 n + 1 φ i ( r i x + q i y ) ∈ Z for real valued functions defined on a linear space V. We derive necessary and sufficient conditions for functions satisfying this condition to be decent in the following sense: there exist functions $$f_i:V\rightarrow {\mathbb {R}}$$ f i : V → R , $$g_i:V\rightarrow {\mathbb {Z}}$$ g i : V → Z such that $$\varphi _i=f_i+g_i$$ φ i = f i + g i , $$(i=0,\dots ,n+1)$$ ( i = 0 , ⋯ , n + 1 ) and $$\sum _{i=0}^{n+1}f_i(r_ix+q_iy)=0$$ ∑ i = 0 n + 1 f i ( r i x + q i y ) = 0 for all $$x, y\in V$$ x , y ∈ V .