• Volume 116, Issue 2

May 2006,   pages  121-255

• 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 as time-frequency analysis or Gabor 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.

• 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.

• Non-Euclidean Visibility Problems

We consider the analog of visibility problems in hyperbolic plane (represented by Poincaré half-plane model $\mathbb{H}$), replacing the standard lattice $\mathbb{Z} × \mathbb{Z}$ by the orbit 𝑧 = 𝑖 under the full modular group $SL_2(\mathbb{Z})$. We prove a visibility criterion and study orchard problem and the cardinality of visible points in large circles.

• Enveloping 𝜎-𝐶*-Algebra of a Smooth Frechet Algebra Crossed Product by $\mathbb{R}, K$-Theory and Differential Structure in 𝐶*-Algebras

Given an 𝑚-tempered strongly continuous action 𝛼 of $\mathbb{R}$ by continuous $∗$-automorphisms of a Frechet $∗$-algebra 𝐴, it is shown that the enveloping 𝜎-𝐶*-algebra $E(S(\mathbb{R},A^∞,𝛼))$ of the smooth Schwartz crossed product $S(\mathbb{R},A^∞,𝛼)$ of the Frechet algebra $A^∞$ of $C^∞$-elements of 𝐴 is isomorphic to the 𝜎-𝐶*-crossed product $C^∗(\mathbb{R}, E(A), 𝛼)$ of the enveloping 𝜎-𝐶*-algebra 𝐸(𝐴) of 𝐴 by the induced action. When 𝐴 is a hermitian $\mathcal{Q}$-algebra, one gets 𝐾-theory isomorphism $R K_∗(S(\mathbb{R},A^∞,𝛼))=K_∗(C^∗(\mathbb{R}, E(A),𝛼)$ for the representable 𝐾-theory of Frechet algebras. An application to the differential structure of a 𝐶*-algebra defined by densely defined differential seminorms is given.

• Quotient Normed Cones

Given a normed cone (𝑋, 𝑝) and a subcone 𝑌, we construct and study the quotient normed cone $(X/Y,\tilde{p})$ generated by 𝑌. In particular we characterize the bicompleteness of $(X/Y,\tilde{p})$ in terms of the bicompleteness of (𝑋, 𝑝), and prove that the dual quotient cone $((X/Y)^∗,\|\cdot\|_{\tilde{p},u})$ can be identified as a distinguished subcone of the dual cone $(X^∗,\|\cdot\|_{\tilde{p},u})$. Furthermore, some parts of the theory are presented in the general setting of the space $CL(X,Y)$ 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.

• Minimal Cuntz–Krieger Dilations and Representations of Cuntz–Krieger Algebras

Given a contractive tuple of Hilbert space operators satisfying certain 𝐴-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 maximal 𝐴-relation piece. We define a maximal piece more generally for a finite set of polynomials in 𝑛 noncommuting variables. We classify all representations of Cuntz–Krieger algebras $\mathcal{O}_A$ obtained from dilations of commuting tuples satisfying 𝐴-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.

• On 𝑝-Quermassintegral Differences Function

In this paper we establish Minkowski inequality and Brunn–Minkowski inequality for 𝑝-quermassintegral differences of convex bodies. Further, we give Minkowski inequality and Brunn–Minkowski inequality for quermassintegral differences of mixed projection bodies.

• Multiplicity of Nontrivial Solutions for Elliptic Equations with Nonsmooth Potential and Resonance at Higher Eigenvalues

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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