On Functional Equations Connected with Quadrature Rules

Koclęga-Kulpa, Barbara; Szostok, Tomasz; Wa̧sowicz, Szymon

doi:10.1515/gmj.2009.725

Cited by 9 publications

(10 citation statements)

References 3 publications

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“…Theorem 3.3. The existence of a nonzero additive solution of (7) implies that there exist a finitely generated subfield K ⊂ C containing α i and β i (i = 1, . .…”

Section: Additive Solutions Of Linear Functional Equationsmentioning

confidence: 99%

“…Equation ( 1) is motivated by quadrature rules of approximate integration. The problem is due to T. Szostok [5], see also [6] and [7]. To formulate the basic preliminary results and facts we need the notion of generalized polynomials.…”

Section: Introductionmentioning

confidence: 99%

“…In what follows we are going to show that spectral analysis can be applied in translation invariant closed linear subspaces (varieties) of additive functions on some finitely generated fields containing the restrictions of the solutions of functional equation (7). Note that the translation invariance is taken with respect to the multiplicative group structure.…”

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confidence: 99%

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

Kiss

Vincze

2017

Aequat. Math.

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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.

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“…Theorem 3.3. The existence of a nonzero additive solution of (7) implies that there exist a finitely generated subfield K ⊂ C containing α i and β i (i = 1, . .…”

Section: Additive Solutions Of Linear Functional Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Kiss

Vincze

2017

Aequat. Math.

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“…The case of p > 2 can be investigated in a similar way because of the inductive argument as follows. The first equation of system (8)…”

Section: The Case Of Higher Transcendence Degreementioning

confidence: 99%

“…where 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].…”

Section: Introductionmentioning

confidence: 99%

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

Kiss¹,

Vincze²

2017

Preprint

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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].

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