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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].

Throughout this paper, we shall suppose that all algebraic number fields, namely, all algebraic extensions over the rational field , are contained in the complex field . Let be the set of all prime numbers. For any algebraic number field , let denote the ideal class group of and, writing for the maximal real subfield of , let denote the kernel of the norm map from to the ideal class group of ; for each , let denote the -class group of , that is, the -primary component of , and let denote the -primary component of . Furthermore, for each , we denote by the ring of -adic integers.

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.

If r is an automorphism and d is a r-derivation of a ring R, then the subring of invariants is the set R(d) fl† r ‘ R r d(r) fl 0·. The main result of this paper is ‘let R be a semiprime ring with an algebraic r-derivation d such that R(d) is central; then R is commutative’. This theorem generalizes results on the invariants of automorphisms and derivations and is proved by reducing down to the special cases of automorphisms and derivations.

In this paper, we continue our investigation of the distribution of the values of 1x 3 1 + 2x 3 2+sx 3 s at integer points when s = 7 and 8. Under suitable conditions on the real algebraic numbers s it is shown that for any real the inequality j 1x 3 1 +sx 3 s - j