. The Balian--Low theorem (BLT) is a key result in time-frequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system fe 2ßimbt g(t \Gamma na)g m;n2Z with ab = 1 forms an orthonormal basis for L 2 (R), then `Z 1 \Gamma1 jt g(t)j 2 dt ' `Z 1 \Gamma1 jfl g(fl)j 2 dfl ' = +1: The BLT was later extended from orthonormal bases to exact frames. This paper presents a tutorial on Gabor systems, the BLT, and related topics, such as the Zak transform and Wilson bases. Because of the fact that (g 0 ) (fl) = 2ßifl g(fl), the role of differentiation in the proof of the BLT is examined carefully. The major new contributions of this paper are the construction of a complete Gabor system of the form fe 2ßibm t g(t \Gamma an )g such that f(an ; bm )g has density strictly less than 1, an Amalgam BLT that provides distinct restrictions on Gabor systems fe 2ßimbt g(t \Gamma na)g that form exact frames, and a new proof of the BLT for exact frame...

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We prove a new regularity result for operators of type L = \Delta +B\Deltar+c, on open sets\Omega ae IR n , provided B :\Omega ! IR n , c :\Omega ! IR satisfy mild integrability conditions. As a consequence we prove that L = \Delta + r log ae \Delta r with domain C 1 0 (IR n ) is essentially self-adjoint on L 2 (IR n ; aedx), if ae 2 H 1;1 loc (IR n ) and r log ae 2 L fl loc (IR n ; dx) for some fl ? n.

The restrictions B s pq and F s pq of the Besov and Triebel--Lizorkin spaces of tempered distributions B s pq (R n ) and F s pq (R n ) to Lipschitz domains\Omega ae R n are studied. For general values of parameters (s 2 R, p ? 0, q ? 0) a "universal" linear bounded extension operator from B s pq and F s pq into the corresponding spaces on R n is constructed. The construction is based on a new variant of the Calder'on reproducing formula with kernels supported in a fixed cone. Explicit characterizations of the elements of B s pq and F s pq in terms of their values in\Omega are also obtained. Introduction The purpose of this paper is to construct a linear operator E which extends functions and distributions from a given Lipschitz domain\Omega ae R n to all of R n and possesses the following property: If a distribution f 2 D 0(\Omega\Gamma can be somehow extended to a tempered distribution g on R n which is "regular" in the sense that g 2 X for some function or...

Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts. 1.

We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate half-spaces are approximate solutions of the isoperimetric problem. 1

Kernel methods have been very popular in the machine learning literature in the last ten years, mainly in the context of Tikhonov regularization algorithms. In this paper we study a coherent Bayesian kernel model based on an integral operator defined as the convolution of a kernel with a signed measure. Priors on the random signed measures correspond to prior distributions on the functions mapped by the integral operator. We study several classes of signed measures and their image mapped by the integral operator. In particular, we identify a general class of measures whose image is dense in the reproducing kernel Hilbert space (RKHS) induced by the kernel. A consequence of this result is a function theoretic foundation for using non-parametric prior specifications in Bayesian modeling, such as Gaussian process and Dirichlet process prior distributions. We discuss the construction of priors on spaces of signed measures using Gaussian and Lévy processes, with the Dirichlet processes being a special case the latter. Computational issues involved with sampling from the posterior distribution are outlined for a univariate regression and a high dimensional

The stationary Schrödinger-Poisson system with a self-consistent effective Kohn-Sham potential is a system of PDEs for the electrostatic potential and the envelopes of wave functions defining the quantum mechanical carrier densities in a semiconductor nanostructure. We regard both Poisson's and Schrödinger's equation with mixed boundary conditions and discontinuous coefficients. Without an exchange correlation potential the Schrödinger-Poisson system is a nonlinear Poisson equation in the dual of a Sobolev space which is determined by the boundary conditions imposed on the electrostatic potential. The nonlinear Poisson operator involved is strongly monotone and boundedly Lipschitz continuous, hence the operator equation has a unique solution. The proof rests upon the following property: the quantum mechanical carrier density operator depending on the potential of the defining Schrödinger operator is antimonotone and boundedly Lipschitz continuous. The solution of the Schrödinger-Poisson system...

Introduction. Set up by Hardy & Littlewood, a theory of rearrangements was popularized by the well-known book [HLP]. Rearrangements of functions are frequently used in real and harmonic analysis, in investigations about singular integrals and function spaces --- see, e.g., [He], [ON], [ONW], [SW]. P'olya & Szego and their followers demonstrated a good many isoperimetric theorems and inequalities by means of rearrangements --- see [PS], a source book on this matter. More recent investigations have shown 178 G. TALENTI that rearrangements of functions fit well also into the theory of elliptic second-order partial differential equations --- see, e.g., [Bae], [Ta3] and the bibliography therein. Several types of rearrangements are known --- presentations are in [Ka] and [Bae]. Here we limit ourselves to rearrangements `a la Hardy & Littlewood. 1.2. Definitions and basic properties. Let G be a measurable subset of R<F

Abstract. We characterize the set of functions which can be approximated by smooth functions and by polynomials with the norm �f � W k, ∞ (w):= k� �f (j) �L ∞ (wj), j=0 for a wide range of (even non-bounded) weights wj’s. We allow a great deal of independence among the weights wj’s.