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

Abstract

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...

On linear functional equations modulo $${\mathbb {Z}}$$

“…Problems connected to class (1), its generalizations and its applications have been studied by several authors during the last more than 100 years. (Recent related results can be found, among others, in [13,[15][16][17]24]. )…”

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

“…In this case we need the so-called spectral synthesis to decide the existence of the nonzero particular solution of the functional equation. On the other hand the spectral synthesis helps us to describe the entire space of the solutions on a large class of finitely generated fields; see [18].…”

“…, p − 1; for the similar trick see [3]. The multivariable form of the conditions will have an important role to clarify the consequences of (18) for the properties of the translated mapping…”

“…In general we also need the application of spectral synthesis in the varieties to give the description of the solution space on finitely generated fields. Since its central objects are the so-called differential operators having a more complicated behavior relative to the automorphism solution, the application of the spectral synthesis is presented in a forthcoming paper [18] as the continuation of the recent one.…”

Computer assisted solution of systems of two variable linear functional equations

“…Early studies of polynomial equations were made by Maurice Fréchet ( [14,15]), while the class above was already considered by W. Harold Wilson ([32]) in the first decades of the previous century. For recent investigations, we refer to the papers [20,[23][24][25]31] and the references therein. Solutions of (1), in the general case when X and Y are certain Abelian groups and the products p i x and q i y in the arguments of the unknown functions are replaced by homomorphisms of X, were determined by László Székelyhidi in [29] (cf., also, [30]).…”