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

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 \cite{KL}, see also G. Kiss and A. Varga \cite{KV}. We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to T. Szostok \cite{KKSZ08}, see also \cite{KKSZ} and \cite{KKSZW}. They are motivated by quadrature rules of approximate integration

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

As a continuation of our previous work \cite{KV2} 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 \cite{KL}. 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 \cc and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration \cite{KKSZ08}, see also \cite{KKSZ} and \cite{KKSZW}

Konform geometria Riemann-Finsler típusú metrikus tereken = Conform geometry of spaces with Riemann-Finsler metrics

Témánk az ún. Finsler-terek konform geometriája, különös tekintettel speciális tértípusok konform ekvivalenciájára. Egy sokaság Finsler-tér, ha az érintővektorok hosszát egy nem feltétlenül belső szorzatból származó funkcionálsereg segítségével mérni tudjuk. Finsler-terek konform ekvivalenciája azt jelenti, hogy a funkcionálok pontonként/érintőterenként egymás skalárszorosai. A legismertebb típus a Berwald-tereké. Egy Wagner-tér pedig mindig konform ekvivalens egy Berwald-térrel. Fő célunk a Wagner-terek belső geometriai jellemzésének a megoldása volt a Matsumoto-féle vannak-e nem triviálisan konform ekvivalens Berwald-terek problémával együtt. Eredményeink szerint két Berwald-tér konform ekvivalenciája mindig triviális hacsak nem Riemann-terekről van szó. Ez azt jelenti, hogy a Wagner-tereket jellemző konform kapcsolat lényegében egyértelműen meghatározott. Végül sikerült a belső geometriai jellemzés problémáját is megoldani egy a konform faktorra felírt parciális differenciálegyenletrendszer segítségével. Az általános elméletet az ún. Randers-terek esetében alkalmaztuk. Itt egy pontonként lineáris taggal deformált Riemann-féle metrikus tenzorral mérünk. Leírtuk azoknak a Riemann-tereknek a lokális struktúráját, melyek megengedik a metrikus tenzor lineáris deformációját úgy, hogy Wagner-teret kapjunk. Ilyen például a konstans negatív görbületű (hiperbolikus) tér. Ezek Wagner-térré deformált osztályát nevezi Szabó Zoltán Bolyai-Lobacsevszkij-Finsler-féle térnek. | The topic is the conformal geometry of Finsler spaces and the conformal equivalence of spaces of special types. Finsler spaces are manifolds equipped with a smooth collection of functionals measuring the length of tangent vectors. The conformal equivalence means that the members of two collections of functionals are homothetic to each other on each tangent space. The class of Berwald spaces is relatively well-known. The Wagner spaces are conformal to Berwald spaces. The aim is an intrinsic characterization via the canonical date of Finsler spaces instead of the extrinsic conformal relation. This is closely related to the problem due to M. Matsumoto: are there conformally equivalent Berwald spaces? Our main result states that the conformal equivalence between two Berwald spaces are always trivial unless they are Riemannian. Consequently the conformal relation of a Finsler space to a Berwald space is essentially unique. The general solution of the intrinsic characterization of Wagner spaces via a system of partial differential equations for the scale function give nice results in case of Randers spaces. The length of tangent vectors is measured by a Riemannian metric perturbed with a 1- form. Our (local) structure theorem characterizes Riemannian spaces admitting a 1-form such that the perturbation results in a Wagner manifold. An important example is the hyperbolic space. In this case the joined Wagner space is called a Bolyai-Lobacsevszkij-Finsler space by Z. Szabó