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193
The Convenient Setting of Global Analysis
, 1997
"... ichor i Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER I Calculus of Smooth Mappings . . . . . . . . . . . . . . . . . . . . 4 1. Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1a. Completeness . . . . . . . . . . . . . . ..."
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Cited by 198 (47 self)
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ichor i Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER I Calculus of Smooth Mappings . . . . . . . . . . . . . . . . . . . . 4 1. Smooth Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1a. Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1b. Smooth Mappings and the Exponential Law . . . . . . . . . . . . . 17 2. The c 1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . 29 3. Uniform Boundedness Principles and Multilinearity . . . . . . . . . . 47 3a. Some Spaces of Smooth Functions . . . . . . . . . . . . . . . . . 59 Historical remarks on the development of smooth calculus . . . . . . . . . 63 CHAPTER II Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . . . 68 5. D
Noncommutative FiniteDimensional Manifolds  I. SPHERICAL MANIFOLDS AND RELATED EXAMPLES
, 2001
"... We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 d ..."
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Cited by 89 (12 self)
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We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic Ktheoretic equations. We find a 3parameter family of deformations of the standard 3sphere S 3 and a corresponding 3parameter deformation of the 4dimensional Euclidean space R 4. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u are isomorphic to the algebras introduced by Sklyanin in connection with the YangBaxter equation. Special values of the deformation parameters do not give rise to Sklyanin algebras and we extract a subclass, the θdeformations, which we generalize in any dimension and various contexts, and study in some details. Here, and
Generalized Model Sets and Dynamical Systems
 CRM Monograph Series
, 1999
"... It is shown that the dynamical systems approach to the diffraction properties of model sets can be generalized to regular model sets in arbitrary sigmacompact Abelian groups with arbitrary locally compact Abelian groups as internal spaces. It is then shown that these regular model sets possess pure ..."
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Cited by 64 (0 self)
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It is shown that the dynamical systems approach to the diffraction properties of model sets can be generalized to regular model sets in arbitrary sigmacompact Abelian groups with arbitrary locally compact Abelian groups as internal spaces. It is then shown that these regular model sets possess pure point diffraction spectra.
A Necessary And Sufficient Condition For The Linear Independence Of The Integer Translates Of A Compactly Supported Distribution
 Constr. Approx
, 1987
"... Given a multivariate compactly supported distribution OE, we derive here a necessary and sufficient condition for the global linear independence of its integer translates. This condition is based on the location of the zeros of b OE = The FourierLaplace transform of OE. The utility of the condition ..."
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Cited by 43 (10 self)
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Given a multivariate compactly supported distribution OE, we derive here a necessary and sufficient condition for the global linear independence of its integer translates. This condition is based on the location of the zeros of b OE = The FourierLaplace transform of OE. The utility of the condition is demonstrated by several examples and applications, showing in particular, that previous results on box splines and exponential box splines can be derived from this condition by a simple combinatorial argument. July, 1987 1980 Mathematics Subject Classification (1985): Primary 41A63, 41A15. Key Words: Box Splines, Exponential Box Splines, Polynomial Box Splines, Integer Translates, Compactly Supported Function, Compactly Supported Distribution, Spectral Analysis, Global Linear Independence, Fourier Transform. 1. Introduction and Statement of Main Results Let E 0 (IR s ) be the space of all sdimensional complex valued distributions of compact support. For each OE 2 E 0 (IR s ) ...
Continuity properties of Schrödinger semigroups with magnetic fields
 Rev. Math. Phys
"... The objects of the present study are oneparameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Katolike conditions. The configuration space ..."
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Cited by 40 (10 self)
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The objects of the present study are oneparameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Katolike conditions. The configuration space is supposed to be an arbitrary open subset of multidimensional Euclidean space; in case that it is a proper subset, the Schrödinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show localnormcontinuity of the semigroup in the potentials. Finally, we prove that the semigroup has a continuous integral kernel given by a Brownianbridge expectation. Altogether, the article is meant to extend some of the results in B. Simon’s landmark paper (Bull. Amer. Math. Soc. (N.S.) 7, 447–526 (1982)) to nonzero vector potentials and more general configuration
Wavelet Methods For Curve Estimation
, 1994
"... The theory of wavelets is a developing branch of mathematics with a wide range of potential applications. Compactly supported wavelets are particularly interesting because of their natural ability to represent data with intrinsically local properties. They are useful for the detection of edges and ..."
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Cited by 37 (7 self)
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The theory of wavelets is a developing branch of mathematics with a wide range of potential applications. Compactly supported wavelets are particularly interesting because of their natural ability to represent data with intrinsically local properties. They are useful for the detection of edges and singularities in image and sound analysis, and for data compression. However, most of the wavelet based procedures currently available do not explicitly account for the presence of noise in the data. A discussion of how this can be done in the setting of some simple nonparametric curve estimation problems is given. Wavelet analogues of some familiar kernel and orthogonal series estimators are introduced and their finite sample and asymptotic properties are studied. We discover that there is a fundamental instability in the asymptotic variance of wavelet estimators caused by the lack of translation invariance of the wavelet transform. This is related to the properties of certain lacunary seq...
Todorov, Rationality of conformally invariant local correlation functions on compactified Minkowski space
"... Minkowski space ..."
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed ..."
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Cited by 28 (10 self)
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We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
TimeFrequency Analysis of Localization Operators
 J. FUNCT. ANAL
, 2002
"... We study a class of pseudodifferential operators known as timefrequency localization operators, AntiWick operators, GaborToeplitz operators or wave packets. Given a symbol a and two windows ' 1 ; ' 2 , we investigate the multilinear mapping from (a; ' 1 ; ' 2 ) 2 S ) to the localization op ..."
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Cited by 26 (16 self)
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We study a class of pseudodifferential operators known as timefrequency localization operators, AntiWick operators, GaborToeplitz operators or wave packets. Given a symbol a and two windows ' 1 ; ' 2 , we investigate the multilinear mapping from (a; ' 1 ; ' 2 ) 2 S ) to the localization operator A a and we give sufficient and necessary conditions for a to be bounded or to belong to a Schatten class. Our results are formulated in terms of timefrequency analysis, in particular we use modulation spaces as appropriate classes for symbols and windows.
The Magnetic Weyl Calculus
, 2004
"... In the presence of a variable magnetic field, the Weyl pseudodifferential calculus must be modified. The usual modification, based on “the minimal coupling principle ” at the level of the classical symbols, does not lead to gauge invariant formulae if the magnetic field is not constant. We present a ..."
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Cited by 21 (18 self)
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In the presence of a variable magnetic field, the Weyl pseudodifferential calculus must be modified. The usual modification, based on “the minimal coupling principle ” at the level of the classical symbols, does not lead to gauge invariant formulae if the magnetic field is not constant. We present a gauge covariant quantization, relying on the magnetic canonical commutation relations. The underlying symbolic calculus is a deformation, defined in terms of the magnetic flux through triangles, of the classical Moyal product.