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Time-variantfilters based on Calderon and Gabor reproducingformulas are important tools in time­ frequencyanalysis.The paper studiesthe behaviorof the eigenvaluesof these filters.Optimal two-sided estimates of the number of eigenvaluescontainedin the interval (151,02), where 0 < 01 < 152 < 1, arc obtained.The estimatescoverlarge classesof localizationdomainsand generatingfunctions. 1. Introduction and statements of the results Calderon- Toeplitz and Gabor- Toeplitz operators arise naturally in two contexts: (i) Toephtz operators on Fock and Bergman spaces of holomorphic functions; (ii) time-variant filters based on Calderon and Gabor reproducing formulas. This paper is concerned with the eigenvalues of a subclass of Calderon- Toeplitz and Gabor- Toeplitz operators which have characteristic functions of bounded domains as symbols. Operators of this class are called time-frequency localiza­ tion operators. The basic idea of functional calculus is that the operators resemble the main algebraic features of their symbols. We consider symbols that are idem­ potent with respect to pointwise multiplication, so it is natural to expect that the corresponding operators are at least approximately idempotent. It is easy to verify that time-frequency localization operators are compact, self-adjoint and bounded by 1. In view of these facts and the above-mentioned correspondence principle, one is inclined to think that localization operators should resemble finite dimensional orthogonal projections. We show that this expectation is correct for Gabor- Toeplitz operators and that it is false for Calderon- Toeplitz operators. We identify the basic geometric features responsible for these two different behaviors. Our principal results are two-sided estimates of the number of eigenvalues inside the plunge region corresponding to 61, (52, where 0 < 61 < (52 < 1. The plunge region consists of the set of indices of the eigenvalues contained inside the open interval (61,62). The eigen­ val ues are ordered non-increasingly. Our work generalizes and improves previ ous results of Daubechies, Paul, Ramanathan and Topiwala [6, 8, 21].

If f is in the Eremenko–Lyubich class B (transcendental entire functions with bounded singular set), then Ω = {z : |f (z)| >R } and f|Ω must satisfy certain simple topological conditions when R is sufficiently large. A model (Ω ,F ) is an open set Ω and a holomorphic function F on Ω that satisfy these same conditions. We show that any model can be approximated by an Eremenko– Lyubich function in a precise sense. In many cases, this allows the construction of functions in B with a desired property to be reduced to the construction of a model with that property, and this is often much easier to do. The singular set of a entire function f is the closure of its critical values and finite asymptotic values and is denoted by S(f ). The Eremenko–Lyubich class B consists of functions such that S(f ) is a bounded set (such functions are also called bounded type). The Speiser class S⊂ B (also called finite type) is those functions for which S(f ) is a finite set. In [9], Eremenko and Lyubich showed that if S(f ) ⊂ DR = {z : |z| R } is a disjoint union of analytic, unbounded, Jordan domains, and that f acts as a covering map f :Ω j →{ x : |z| >R } on each component Ωj of Ω. Building examples where Ω has a certain geometry is important for applications to dynamics. We would like to start with a model, that is, a choice of Ω and a covering map f :Ω →{ |z| > 1} and ask whether f can be approximated by an entire function F in B or S. In this paper, we deal with approximation by functions in B. It turns out that if Ω satisfies some obviously necessary topological conditions, then the approximation by Eremenko–Lyubich functions is always possible in a sense strong enough to imply that the functions f and F have the same dynamical behavior on their Julia sets. This allows us to build entire functions in B with certain behaviors by simply exhibiting a model with that behavior (this is often much easier to do). In [4], we deal with the analogous question for the Speiser class; again the approximation is always possible, but in a slightly weaker sense (dynamically, given any model we can build a function in the Speiser class that has the model’s dynamics on some subset of its Julia set). In the next few paragraphs, we introduce some notation to make these remarks more precise. Suppose that Ω = � j Ωj is a disjoint union of unbounded simply connected domains

We revise the concept of compact tripotent in the bidual space of a JB*-triple. This concept was introduced by Edwards and Ruttimann generalizing the ideas developed by Akemann for compact projections in the bidual of a C*-algebra. We also obtain some characterizations of weak compactness in the dual space of a JC*-triple, showing that a bounded subset in the dual space of a JC*-triple is relatively weakly compact if and only if its restriction to any abelian maximal subtriple C is relatively weakly compact in the dual of C. This generalizes a very useful result by Pfitzner in the setting of C*-algebras. As a consequence we obtain a Dieudonnet heorem for JC*-triples which generalizes the one obtained by Brooks, Saito and Wright for C*-algebras. One of the most celebrated and useful results characterizing weakly compact subsets in the dual space of a C*-algebra is due to Pfitzner, who established that weak com- pactness in the dual space of a C*-algebra is commutatively determined (see (30)). More concretely, Pfitzner shows, in a 'tour de force', that if K is a bounded subset in the dual space of a C*-algebra A ,t henK is relatively weakly compact if and only if the restriction of K to each maximal abelian subalgebra of A is relatively weakly compact. This result has many important consequences, one of the most interesting being that every C*-algebra satisfies property (V) of Pelczynski. Pfitzner's result is the latest advance in the study of weak compactness in the dual space of a C*-algebra developed by Takesaki (33), Akemann (1), Akemann, Dodds and Gamlen (3), Saito( 32) and Jarchow (22, 23). In the more general setting of dual spaces of JB*-triples the study of weak compactness has been developed by Chu and Iochum (12) and by Peralta and Rodr´ iguez-Palacios (28, 29). However, all the results concerning weak compactness in the dual space of a JB*-triple give characterizations in terms of the abelian subtriples of its bidual instead of the abelian subtriples of the JB*-triple itself. The question clearly is whether a bounded subset in the dual space of a JB*-triple, E, is relatively weakly compact whenever its restriction to any abelian subtriple of E is. In the main result of this paper we show that weak compactness in the dual space of a JC*-triple is commutatively determined, by showing that a bounded subset K in the dual space of a JC*-triple E is relatively weakly compact if and only if the restriction of K to each separable abelian subtriple of E also is relatively weakly compact (see Theorem 3.5).

The paper investigates the vectorial Dirichlet problem defined by Here is an open bounded subset of with boundary , and () denote the singular values of the gradient . The existence of solutions is established under one of the following assumptions: is continuous on and locally contractive on , or is contractive on . This extends a result due to Dacorogna and Marcellini. The approach is based on the Baire category method developed earlier by the authors.

The paper computes the Hausdorff dimension of sets of real numbers which are close to infinitely many real algebraic integers of bounded degree. It also investigates the distribution of real algebraic integers of bounded degree, which are proved to be evenly spaced.

This paper brings together two apparently unrelated results about locally compact groups G by giving them a common proof. The first concerns the number of topologically left invariant means on L ∞ ( G ), while the second states that the topological centre of the largest semigroup compactification of G is simply G itself. On the way, we introduce as vital tools some new compactifications of the half line ([0, ∞), +), we produce a right invariant pseudometric on a compactly generated G for which the bounded sets are precisely the relatively compact sets, and we receive striking confirmation that some algebraic properties of semigroups can be transferred by maps which are quite far from being homomorphisms.