Kutatások a diadikus harmonikus analízis körében = Research in dyadic harmonic analysis

A pályázat keretében írott cikkek között számosban foglalkoztam egy és kétváltozós integrálható függvények logaritmikus közepeinek konvergenciájával. Többek között vizsgáltuk, hogy mi a legbővebb norma konvergencia tér. A kutatási időszak fő eredménye: Gát, G.: Pointwise convergence of cone-like restricted two-dimensional (C,1) means of trigonometric Fourier series, Journal of Approximation Theory, 149 (1) (2007), 74-102. Marcinkiewicz és Zygmund 1939-ben igazolta kétváltozós trigonometrikus Fourier sorok Fejér közepeivel kapcsolatban, hogy a integrálható függvények kétdimenziós Fejér közepei majdnem mindenütt a függvényhez tartanak, hacsak az közepek indexei úgy tartanak végtelenbe, hogy a hányadosuk korlátos, azaz egy egyenes köré húzott kúpban maradnak. A nevezett cikkben igazoltam, hogy, ha az egyenest helyettesítjük egy függvény görbéjével, azaz egy ''görbe köré húzott kúpban maradnak az indexek'', akkor is igaz marad a majdnem mindenütti konvergencia. Továbbá, ha a "kúp jellegű" halmaz "végtelenül bővül", akkor a tétel már nem fog teljesülni. | Among the papers written in the project I discussed the convergence of logarithmic means of one and two dimensional functions in several papers. Among others, we determinded the largest norm convergence space. The main result of the project is: Gát, G.: Pointwise convergence of cone-like restricted two-dimensional (C,1) means of trigonometric Fourier series, Journal of Approximation Theory, 149 (1) (2007), 74-102. In 1939 Marcinkiewicz and Zygmund proved with respect to the Fejér means of the trigonometric Fourier series of two variable integrable functions that if the ratio of the indices of the means remain bounded as they tend to infinity (in other words, they remain in some positive cone around of the identical function), then the Fejér means converge to the function almost everywhere. In my paper above I verified the same result for a more general case. That is, the identical function can be substituted by an "arbitrary" function. That is, the set of indices remain in a "cone-like" set ("a cone around a curve"). Moreover, if the "cone-like set" enlarges "infinitely", then the theorem fails to hold

L_(p)-error estimates for radial basis function interpolation on the sphere

In this paper we review the variational approach to radial basis function interpolation on the sphere and establish new Lp-error bounds, for p[1,∞]. These bounds are given in terms of a measure of the density of the interpolation points, the dimension of the sphere and the smoothness of the underlying basis function

A Duchon framework for the sphere

In his fundamental paper (RAIRO Anal. Numer. 12 (1978) 325) Duchon presented a strategy for analysing the accuracy of surface spline interpolants to sufficiently smooth target functions. In the mid-1990s Duchon's strategy was revisited by Light and Wayne (J. Approx. Theory 92 (1992) 245) and Wendland (in: A. Le Méhauté, C. Rabut, L.L. Schumaker (Eds.), Surface Fitting and Multiresolution Methods, Vanderbilt Univ. Press, Nashville, 1997, pp. 337–344), who successfully used it to provide useful error estimates for radial basis function interpolation in Euclidean space. A relatively new and closely related area of interest is to investigate how well radial basis functions interpolate data which are restricted to the surface of a unit sphere. In this paper we present a modified version Duchon's strategy for the sphere; this is used in our follow up paper (Lp-error estimates for radial basis function interpolation on the sphere, preprint, 2002) to provide new Lp error estimates (p[1,∞]) for radial basis function interpolation on the sphere

Generalized Contraction and Invariant Approximation Resultson Nonconvex Subsets of Normed Spaces

Wardowski (2012) introduced a new type of contractive mapping and proved a fixed point result in complete metric spaces as a generalization of Banach contraction principle. In this paper, we introduce a notion of generalized F-contraction mappings which is used to prove a fixed point result for generalized nonexpansive mappings on star-shaped subsets of normed linear spaces. Some theorems on invariant approximations in normed linear spaces are also deduced. Our results extend, unify, and generalize comparable results in the literature.The authors are very grateful to the referees for their valuable comments and suggestions, and, in particular, to one of them for calling our attention on the crucial fact stated in the first part of Remark 5 and for the elegant reformulation of Theorem 13 stated in Remark 14. Salvador Romaguera acknowledges the support of the Universitat Politecnica de Valencia, Grant PAID-06-12-SP20120471.Abbas, M.; Ali, B.; Romaguera Bonilla, S. (2014). Generalized Contraction and Invariant Approximation Resultson Nonconvex Subsets of Normed Spaces. Abstract and Applied Analysis. 2014:1-5. https://doi.org/10.1155/2014/391952S152014Banach, S. (1922). Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae, 3, 133-181. doi:10.4064/fm-3-1-133-181Arandjelović, I., Kadelburg, Z., & Radenović, S. (2011). Boyd–Wong-type common fixed point results in cone metric spaces. Applied Mathematics and Computation, 217(17), 7167-7171. doi:10.1016/j.amc.2011.01.113Boyd, D. W., & Wong, J. S. W. (1969). On nonlinear contractions. Proceedings of the American Mathematical Society, 20(2), 458-458. doi:10.1090/s0002-9939-1969-0239559-9Huang, L.-G., & Zhang, X. (2007). Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications, 332(2), 1468-1476. doi:10.1016/j.jmaa.2005.03.087Rakotch, E. (1962). A note on contractive mappings. Proceedings of the American Mathematical Society, 13(3), 459-459. doi:10.1090/s0002-9939-1962-0148046-1Tarafdar, E. (1974). An approach to fixed-point theorems on uniform spaces. Transactions of the American Mathematical Society, 191, 209-209. doi:10.1090/s0002-9947-1974-0362283-5Dix, J. G., & Karakostas, G. L. (2009). A fixed-point theorem for S-type operators on Banach spaces and its applications to boundary-value problems. Nonlinear Analysis: Theory, Methods & Applications, 71(9), 3872-3880. doi:10.1016/j.na.2009.02.057Latrach, K., Aziz Taoudi, M., & Zeghal, A. (2006). Some fixed point theorems of the Schauder and the Krasnosel’skii type and application to nonlinear transport equations. Journal of Differential Equations, 221(1), 256-271. doi:10.1016/j.jde.2005.04.010Meinardus, G. (1963). Invarianz bei linearen Approximationen. Archive for Rational Mechanics and Analysis, 14(1), 301-303. doi:10.1007/bf00250708Habiniak, L. (1989). Fixed point theorems and invariant approximations. Journal of Approximation Theory, 56(3), 241-244. doi:10.1016/0021-9045(89)90113-5Hicks, T. ., & Humphries, M. . (1982). A note on fixed-point theorems. Journal of Approximation Theory, 34(3), 221-225. doi:10.1016/0021-9045(82)90012-0Singh, S. . (1979). An application of a fixed-point theorem to approximation theory. Journal of Approximation Theory, 25(1), 89-90. doi:10.1016/0021-9045(79)90036-4Subrahmanyam, P. . (1977). An application of a fixed point theorem to best approximation. Journal of Approximation Theory, 20(2), 165-172. doi:10.1016/0021-9045(77)90070-3Wardowski, D. (2012). Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory and Applications, 2012(1). doi:10.1186/1687-1812-2012-94Abbas, M., Ali, B., & Romaguera, S. (2013). Fixed and periodic points of generalized contractions in metric spaces. Fixed Point Theory and Applications, 2013(1), 243. doi:10.1186/1687-1812-2013-24