Volume 116, Issue 2
May 2006, pages 121-255
pp 121-136 May 2006 Invited Article
The abstruse meets the applicable: Some aspects of time-frequency analysis
The area of Fourier analysis connected to signal processing theory has undergone a rapid development in the last two decades. The aspect of this development that has received the most publicity is the theory of wavelets and their relatives, which involves expansions in terms of sets of functions generated from a single function by translations and dilations. However, there has also been much progress in the related area known astime-frequency analysis orGabor analysis, which involves expansions in terms of sets of functions generated from a single function by translations and modulations. In this area there are some questions of a concrete and practical nature whose study reveals connections with aspects of harmonic and functional analysis that were previously considered quite pure and perhaps rather exotic. In this expository paper, I give a survey of some of these interactions between the abstruse and the applicable. It is based on the thematic lectures which I gave at the Ninth Discussion Meeting on Harmonic Analysis at the Harish-Chandra Research Institute in Allahabad in October 2005.
pp 137-146 May 2006 Regular Articles
On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis
There has been a surge of interest of late in an old result of Nyman and Beurling giving a Hilbert space formulation of the Riemann hypothesis. Many authors have contributed to this circle of ideas, culminating in a beautiful refinement due to Baez-Duarte. The purpose of this little survey is to dis-entangle the resulting web of complications, and reveal the essential simplicity of the main results.
pp 147-160 May 2006 Regular Articles
Non-Euclidean visibility problems
We consider the analog of visibility problems in hyperbolic plane (represented by Poincaré half-plane model ℍ), replacing the standard lattice ℤ × ℤ by the orbitz = i under the full modular group SL_{2}(ℤ). We prove a visibility criterion and study orchard problem and the cardinality of visible points in large circles.
pp 161-173 May 2006 Regular Articles
Given anm-tempered strongly continuous action α of ℝ by continuous^{*}-automorphisms of a Frechet^{*}-algebraA, it is shown that the enveloping ↡-C^{*}-algebraE(S(ℝ, A^{∞}, α)) of the smooth Schwartz crossed productS(ℝ,A^{∞}, α) of the Frechet algebra A^{∞} of C^{∞}-elements ofA is isomorphic to the Σ-C^{*}-crossed productC^{*}(ℝ,E(A), α) of the enveloping Σ-C^{*}-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK_{*}(S(ℝ, A^{∞}, α)) =K^{*}(C^{*}(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC^{*}-algebra defined by densely defined differential seminorms is given.
pp 175-191 May 2006 Regular Articles
Given a normed cone (X, p) and a subconeY, we construct and study the quotient normed cone (X/Y,p) generated byY. In particular we characterize the bicompleteness of (X/Y, ‖·‖^{p},p) in terms of the bicompleteness of (X, p), and prove that the dual quotient cone ((X/Y)^{*}, ¦¦ · ‖·‖^{p},p) can be identified as a distinguished subcone of the dual cone (X^{*}, ¦¦ · ¦¦_{p, u}). Furthermore, some parts of the theory are presented in the general setting of the spaceCL(X, Y) of all continuous linear mappings from a normed cone (X, p) to a normed cone (Y, q), extending several well-known results related to open continuous linear mappings between normed linear spaces.
pp 193-220 May 2006 Regular Articles
Minimal Cuntz-Krieger dilations and representations of Cuntz-Krieger algebras
B V Rajarama Bhat Santanu Dey Joachim Zacharias
Given a contractive tuple of Hilbert space operators satisfying certainA-relations we show that there exists a unique minimal dilation to generators of Cuntz-Krieger algebras or its extension by compact operators. This Cuntz-Krieger dilation can be obtained from the classical minimal isometric dilation as a certain maximalA-relation piece. We define a maximal piece more generally for a finite set of polynomials inn noncommuting variables. We classify all representations of Cuntz-Krieger algebrasO_{A} obtained from dilations of commuting tuples satisfyingA-relations. The universal properties of the minimal Cuntz-Krieger dilation and the WOT-closed algebra generated by it is studied in terms of invariant subspaces.
pp 221-231 May 2006 Regular Articles
Onp-quermassintegral differences function
In this paper we establish Minkowski inequality and Brunn-Minkowski inequality forp-quermassintegral differences of convex bodies. Further, we give Minkowski inequality and Brunn-Minkowski inequality for quermassintegral differences of mixed projection bodies.
pp 233-255 May 2006 Regular Articles
Leszek Gasiński Dumitru Motreanu Nikolaos S Papageorgiou
We consider a semilinear elliptic equation with a nonsmooth, locally Lipschitz potential function (hemivariational inequality). Our hypotheses permit double resonance at infinity and at zero (double-double resonance situation). Our approach is based on the nonsmooth critical point theory for locally Lipschitz functionals and uses an abstract multiplicity result under local linking and an extension of the Castro-Lazer-Thews reduction method to a nonsmooth setting, which we develop here using tools from nonsmooth analysis.
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