Existence of nontrivial solutions of linear functional equations

Kiss, Gergely; Varga, Adrienn

doi:10.1007/s00010-013-0212-z

Cited by 14 publications

(24 citation statements)

References 8 publications

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“…These applications are, in a way, the continuations and completions of the groundbreaking observations, results and examples given in the papers [3] and [2].…”

Section: Introductionmentioning

confidence: 88%

“…This is not true for a general equation of the form (1). As it was noted in [3], the following condition implies translation invariance:…”

Section: Introductionmentioning

confidence: 94%

“…Then it follows from Theorem 7.4 that there are automorphisms φ, ψ, χ of C such that φ · ψ · χ is a solution. Then, by [3,Theorem 3.8], the functions φ · ψ, φ · χ, ψ · χ are also solutions. As we saw above, this implies that φ = ψ = χ on K, and thus φ 3 is a solution.…”

Section: An Example and Concluding Remarksmentioning

confidence: 99%

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Linear functional equations, differential operators and spectral synthesis

Kiss

Laczkovich

2014

Aequat. Math.

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We investigate the functional equation n i=1 a i f (b i x + c i y) = 0, where a i , b i , c i ∈ C, and the unknown function f is defined on the field K = Q(b 1 , . . . , b n , c 1 , . . . , c n ). (It is easy to see that every solution on K can be extended to C as a solution.) Let S 1 denote the set of additive solutions defined on K. We prove that S 1 is spanned by S 1 ∩ D, where D is the set of the functions φ • D, where φ is a field automorphism of C and D is a differential operator on K. We say that the equation We show that S is spanned by S ∩ A, where A is the algebra generated by D. This implies that if S is translation invariant, then spectral synthesis holds in S. The main ingredient of the proof is the observation that if V is a variety on the Abelian group (K * ) k under multiplication, and every function F ∈ V is k-additive on K k , then spectral synthesis holds in V .We give several applications, and describe the set of solutions of equations having some special properties (e.g. having algebraic coefficients etc.).

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“…These applications are, in a way, the continuations and completions of the groundbreaking observations, results and examples given in the papers [3] and [2].…”

Section: Introductionmentioning

confidence: 88%

“…This is not true for a general equation of the form (1). As it was noted in [3], the following condition implies translation invariance:…”

Section: Introductionmentioning

confidence: 94%

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Linear functional equations, differential operators and spectral synthesis

Kiss

Laczkovich

2014

Aequat. Math.

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“…M. Laczkovich [11], A. Varga and G. Kiss [9] clarified that the existence of injective homomorphisms satisfying some special relationships is a necessary and sufficient condition for the existence of a non-trivial k-additive solution. The proof uses the theorem due to M. Laczkovich and G. Székelyhi-di [8].…”

Section: Theorem 13 [4]mentioning

confidence: 99%

Algebraic methods for the solution of linear functional equations

2015

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belongs to the class of linear functional equations. The solutions form a linear space with respect to the usual pointwise operations. According to the classical results of the theory they must be generalized polynomials. New investigations have been started few years ago. They clarified that the existence of non-trivial solutions depends on the algebraic properties of some related families of parameters. The problem is to find the necessary and sufficient conditions for the existence of non-trivial solutions in terms of these kinds of properties. One of the most earlier results is due to Z. Daróczy [1]. It can be considered as the solution of the problem in case of n = 2. We are going to take more steps forward by solving the problem in case of n = 3.

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“…Using some recent results [5], see also [2] and [4], spectral synthesis and spectral analysis hold in some related varieties of the functional equation. The spectral analysis allows us to choose a special kind of solution called exponentials.…”

Section: Introduction Consider Functional Equationmentioning

confidence: 99%

Nontrivial solutions of linear functional equations: methods and examples

Varga¹,

Vincze²

2015

OpMath

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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.

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Existence of nontrivial solutions of linear functional equations

Abstract: Abstract. In this paper we study functional equation n

Cited by 14 publications

References 8 publications

Linear functional equations, differential operators and spectral synthesis

Linear functional equations, differential operators and spectral synthesis

Algebraic methods for the solution of linear functional equations

Nontrivial solutions of linear functional equations: methods and examples

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