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D'Alembert's functional equation on groups

Henrik Stetkær (2013)

Banach Center Publications

Given a (not necessarily unitary) character μ:G → (ℂ∖0,·) of a group G we find the solutions g: G → ℂ of the following version of d’Alembert’s functional equation g ( x y ) + μ ( y ) g ( x y - 1 ) = 2 g ( x ) g ( y ) , x,y ∈ G. (*) The classical equation is the case of μ = 1 and G = ℝ. The non-zero solutions of (*) are the normalized traces of certain representations of G on ℂ². Davison proved this via his work [20] on the pre-d’Alembert functional equation on monoids. The present paper presents a detailed exposition of these results working directly...

Defining complete and observable chaos

Víctor Jiménez López (1996)

Annales Polonici Mathematici

For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x,y) ∈ I² for which l i m i n f n → ∞ | f n ( x ) - f n ( y ) | = 0 and l i m s u p n → ∞ | f n ( x ) - f n ( y ) | > 0 . We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ² denotes the Lebesgue measure on the square and Ch(f) is the set of points (x,y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ²(C(f)) > 0 need not imply λ²(Ch(f)) > 0. We use these results to propose some plausible definitions...

Determinant criteria of solvability

V. S. Kalnitsky (2013)

Banach Center Publications

In the paper [3] the determinant criterion of solvability for the Kuczma equation [4] was given. This criterion appeared in the natural way as barycenter of some mass system. It turned out that determinants do appear in many different situations as solvability criteria. The present article is aimed to review the mostly classical results in the theory of functional equations from this point of view. We begin with classical results of the linear functional equations and the determinant equations solved...

Développements asymptotiques q -Gevrey et séries G q -sommables

Changgui Zhang (1999)

Annales de l'institut Fourier

Nous donnons une version q -analogue de l’asymptotique Gevrey et de la sommabilité de Borel, dues respectivement à G. Watson et E. Borel et systématiquement développées depuis une quinzaine d’années par J.-P. Ramis, Y. Sibuya, etc. Le but de ces auteurs était l’étude des équations différentielles dans le champ complexe. De même notre but est l’étude des équations aux q -différences dans le champ complexe, dans la ligne de G.D. Birkhoff et W.J. Trjitzinsky.Plus précisément, nous introduisons une nouvelle...

Differentiable solutions for a class of functional equations

V. Murugan, P. V. Subrahmanyam (2007)

Annales Polonici Mathematici

We give a set of sufficient conditions for the existence of differentiable solutions for a functional equation involving a series of iterates, using a method different from that of Baker and Zhang [Ann. Polon. Math. 73 (2000)].

Currently displaying 301 – 320 of 1591