Browsing by Subject "Dirichlet"

Now showing 1 - 6 of 6

Open Access
Aspects of multivariable operator theory on weighted symmetric Fock spaces
(World Scientific Publishing, 2014) Kaptanoğlu, H. T.
We obtain all Dirichlet spaces Fq, q ∈ ℝ, of holomorphic functions on the unit ball of ℂN as weighted symmetric Fock spaces over ℂN. We develop the basics of operator theory on these spaces related to shift operators. We do a complete analysis of the effect of q ∈ ℝ in the topics we touch upon. Our approach is concrete and explicit. We use more function theory and reduce many proofs to checking results on diagonal operators on the Fq. We pick out the analytic Hilbert modules from among the Fq. We obtain von Neumann inequalities for row contractions on a Hilbert space with respect to each Fq. We determine the commutants and investigate the almost normality of the shift operators. We prove that the C∗-algebras generated by the shift operators on the Fq fit in exact sequences that are in the same Ext class. We identify the groups K0 and K1 of the Toeplitz algebras on the Fq arising in K-theory. Radial differential operators are prominent throughout. Some of our results, especially those pertaining to lower negative values of q, are new even for N = 1. Many of our results are valid in the more general weighted symmetric Fock spaces Fb that depend on a weight sequence b. © World Scientific Publishing Company.
Open Access
Carleson measures for Besov spaces on the ball with applications
(Academic Press, 2007) Kaptanoǧlu, Hakkı Turgay
Carleson and vanishing Carleson measures for Besov spaces on the unit ball of CN are characterized in terms of Berezin transforms and Bergman-metric balls. The measures are defined via natural imbeddings of Besov spaces into Lebesgue classes by certain combinations of radial derivatives. Membership in Schatten classes of the imbeddings is considered too. Some Carleson measures are not finite, but the results extend and provide new insight to those known for weighted Bergman spaces. Special cases pertain to Arveson and Dirichlet spaces, and a unified view with the usual Hardy-space Carleson measures is presented by letting the order of the radial derivatives tend to 0. Weak convergence in Besov spaces is also characterized, and weakly 0-convergent families are exhibited. Applications are given to separated sequences, operators of Forelli-Rudin type, gap series, characterizations of weighted Bloch, Lipschitz, and growth spaces, inequalities of Fejér-Riesz and Hardy-Littlewood type, and integration operators of Cesàro type.
Open Access
Novelty detection for topic tracking
(John Wiley & Sons, Inc., 2012) Aksoy, C.; Can, F.; Kocberber, S.
Multisource web news portals provide various advantages such as richness in news content and an opportunity to follow developments from different perspectives. However, in such environments, news variety and quantity can have an overwhelming effect. New-event detection and topic-tracking studies address this problem. They examine news streams and organize stories according to their events; however, several tracking stories of an event/topic may contain no new information (i.e., no novelty). We study the novelty detection (ND) problem on the tracking news of a particular topic. For this purpose, we build a Turkish ND test collection called BilNov-2005 and propose the usage of three ND methods: a cosine-similarity (CS)-based method, a language-model (LM)-based method, and a cover-coefficient (CC)-based method. For the LM-based ND method, we show that a simpler smoothing approach, Dirichlet smoothing, can have similar performance to a more complex smoothing approach, Shrinkage smoothing. We introduce a baseline that shows the performance of a system with random novelty decisions. In addition, a category-based threshold learning method is used for the first time in ND literature. The experimental results show that the LM-based ND method significantly outperforms the CS- and CC-based methods, and categorybased threshold learning achieves promising results when compared to general threshold learning. © 2011 ASIS&T.
Open Access
Reproducing Kernels and Radial Differential Operators for Holomorphic and Harmonic Besov Spaces on Unit Balls: a Unified View
(Springer, 2010-07-28) Kaptanoğlu, T.
Open Access
Reproducing kernels of harmonic Besov spaces on the ball
(Elsevier, 2009-07) Gergun, S.; Kaptanoglu, H. T.; Ureyen, A. E.
Besov spaces of harmonic functions on the unit ball of Rn are defined by requiring sufficiently high-order derivatives of functions lie in harmonic Bergman spaces. We compute the reproducing kernels of those Besov spaces that are Hilbert spaces. The kernels turn out to be weighted infinite sums of zonal harmonics and natural radial fractional derivatives of the Poisson kernel. To cite this article: S. Gergün et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). © 2009 Académie des sciences.
Open Access
Toeplitz operators on arveson and dirichlet spaces
(Birkhaeuser Science, 2007) Alpay, D.; Kaptanoǧlu, H. T.
We define Toeplitz operators on all Dirichlet spaces on the unit ball of CN and develop their basic properties. We characterize bounded, compact, and Schatten-class Toeplitz operators with positive symbols in terms of Carleson measures and Berezin transforms. Our results naturally extend those known for weighted Bergman spaces, a special case applies to the Arveson space, and we recover the classical Hardy-space Toeplitz operators in a limiting case; thus we unify the theory of Toeplitz operators on all these spaces. We apply our operators to a characterization of bounded, compact, and Schatten-class weighted composition operators on weighted Bergman spaces of the ball. We lastly investigate some connections between Toeplitz and shift operators. © Birkhäuser Verlag Basel/Switzerland 2007.