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Continuity properties for modulation spaces, with applications to pseudo-differential calculus-I

JOURNAL OF FUNCTIONAL ANALYSIS 2004年第2期207卷 399-429页

作者： Toft, J Univ Karlskrona Dept Math Karlskrona Sweden

Let M-p,M- q denote the modulation space with parameters p, q is an element of [1, infinity]. If 1/p(1) + 1/p(2) = 1 + 1/p(0) and 1/q(1) + 1/q(2) = 1/q(0), then it is proved that M-p1,M- q1 * M-p2,M- q2 subset of M-p0,M- q0. The result is used to get inclusions between modulation spaces, Besov spaces and Schatten classes in calculus of psido (pseudo-differential operators), and to extend the definition of Toeplitz operators. We also discuss continuity of ambiguity functions and psido in the framework of modulation spaces. (C) 2003 Elsevier Inc. All rights reserved.

关键词： modulation spaces pseudo-differential operators embeddings convolution Toeplitz operators Schatten classes

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Bilinear pseudodifferential operators on modulation spaces

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2004年第3期10卷 301-313页

作者： Bényi, A Okoudjou, K Univ Massachusetts Lederle Grad Res Tower Dept Math & Stat Amherst MA 01003 USA Georgia Inst Technol Sch Math Atlanta GA 30332 USA

We use the theory of Gabor frames to prove the boundedness of bilinear pseudodifferential operators on products of modulation spaces. In particular, we show that bilinear pseudodifferential operators corresponding to ... 详细信息

关键词： bilinear operators Gabor frames modulation spaces moderate weights pseudodifferential operators sequence spaces short-time Fourier transform time-frequency analysis weighted spaces

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Symmetrically localized frames and the removal of subsets of positive density

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2007年第2期326卷 1225-1235页

作者： Futamura, Fumiko Vanderbilt Univ Dept Math Nashville TN 37240 USA

The theorems of Balan, Casazza, Heil, and Landau concerning the removal of sets of positive density from frames with positive excess are extended using a more general, symmetric concept of localization of frames. (c) ... 详细信息

关键词： frames localized frames density excess overcompleteness modulation spaces Gabor systems Weyl-Heisenberg systems

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History and evolution of the Density Theorem for Gabor Frames

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2007年第2期13卷 113-166页

作者： Heil, Christopher Georgia Inst Technol Sch Math Atlanta GA 30332 USA

The Density Theorem for Gabor Frames is one of the fundamental results of time-frequency analysis. This expository survey attempts to reconstruct the long and very involved history of this theorem and to present its con text and evolution, from the one-dimensional rectangular lattice setting, to arbitrary lattices in higher dimensions, to irregular Gabor frames, and most recently beyond the setting of Gabor frames to abstract localized frames. Related fundamental principles in Gabor analysis are also surveyed, including the Wexler-Raz biorthogonality relations, the Duality Principle, the Balian-Low Theorem, the Walnut and Janssen representations, and the Homogeneous Approximation Property. An extended bibliography is included.

关键词： Balian-Low Theorem complete sequence density theorem duality principle frames Gabor system homogeneous approximation property Janssen representation localized frames modulation spaces Riesz bases Schauder bases time-frequency analysis von Neumann algebras Walnut representation Wexler-Raz biorthogonality relations Weyl-Heisenberg system

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The global Cauchy problem for the NLS and NLKG with small rough data

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JOURNAL OF DIFFERENTIAL EQUATIONS 2007年第1期232卷 36-73页

作者： Wang Baoxiang Hudzik, Henryk Peking Univ Sch Math Sci LMAM Beijing 100871 Peoples R China Adam Mickiewicz Univ Fac Math & Comp Sci PL-61614 Poznan Poland

By using the unit-cube decomposition to the frequency spaces, we study the Cauchy problem for the nonlinear Schrodinger equation and the nonlinear Klein-Gordon equation. Some global well posedness results are obtained... 详细信息

关键词： nonlinear Schrodinger equation nonlinear Klein-Gordon equation Cauchy problem modulation spaces M-p,q(s) small rough data

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Inversion of the short-time fourier transform using Riemannian sums

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2007年第3期13卷 357-368页

作者： Weisz, Ferenc Eotvos Lorand Univ Dept Numer Anal H-1117 Budapest Hungary

The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L-2 and modulation spaces. In the present note the Riemannian sums of the inverse short-time Fourier transform are investigated. Under some conditions oil the window functions we prove that the Riemannian sums converge to f in the modulation spaces and in Wiener amalgam norms, hence also in the L-p sense.

关键词： wiener amalgam spaces modulation spaces short-time fourier transform time-frequency analysis feichfinger's and wiener's algebra riemannian sums

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TIME-FREQUENCY LOCALIZATIONS FOR modulation spaces ON LOCALLY COMPACT ABELIAN GROUPS

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International Journal of Wavelets, Multiresolution and Information Processing 2004年第2期2卷 149-163页

作者： S. S. PANDEY Department of Mathematics R. D. University Jabalpur 482001 India

In the present paper we define weighted modulation spaces on a LCA group with respect to a window function drawn from a suitable Banach space of test functions and prove a theorem to establish uncertainty principle for these modulation spaces. Also, using the concept of Zak transform, we generalize an earlier result of Heil (1990) on the Balian–Low theorem for the Wiener amalgam space . Our theorems include the corresponding results on Euclidean spaces as particular cases.

关键词： Gabor wavelet transform modulation spaces Feichtinger algebra uncertainty principle Wiener amalgams and Balian–Low theorem

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Boundedness of pseudodifferential operators on modulation spaces

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2003年第1期284卷 389-396页

作者： Czaja, W Univ Wroclaw Inst Math PL-50384 Wroclaw Poland

We study classes of pseudodifferential operators which are bounded on large collections of modulation spaces. The conditions on the operators are stated in terms of the L-p,L-q estimates for the continuous Gabor transforms of their symbols. In particular, we show how these classes are related to the class of operators of Grochenig and Heil, which is bounded on all modulation spaces. (C) 2003 Elsevier Inc. All rights reserved.

关键词： Gabor transform short-time Fourier transform modulation spaces pseudodifferential operators

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Density, overcompleteness, and localization of frames

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ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY 2006年第10期12卷 71-86页

作者： Balan, Radu Casazza, Peter G. Heil, Christopher Landau, Zeph Siemens Corp Res Princeton NJ 08540 USA Univ Missouri Dept Math Columbia MO 65211 USA Georgia Inst Technol Sch Math Atlanta GA 30332 USA CUNY City Coll Dept Math New York NY 10031 USA

This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames F = {fi}(i is an element of I) and epsilon = {e(j)} (j is an element of G) (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E via a map a: I. G. A fundamental set of equalities are shown between three seemingly unrelated quantities: the relative measure of F, the relative measure of E-both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements-and the density of the set a( I) in G. Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. These abstract results yield an array of new implications for irregular Gabor frames. Various Nyquist density results for Gabor frames are recovered as special cases, but in the process both their meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally, these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of concentration in the time-frequency plane.

关键词： density excess frames Gabor systems modulation spaces overcompleteness Riesz bases wavelets Weyl-Heisenberg systems

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Density, overcompleteness, and localization of frames. I. Theory

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2006年第2期12卷 105-143页

Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative framework for describing the overcompleteness of frames. It introduces notions of localization and approximation between two frames F = {f(i)}(i is an element of l) and epsilon = {e(j)} (j is an element of G) (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of epsilon via a map a: I -> G. A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of F, the relative measure of epsilon - both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements - and the density of the set a(I) in G. Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent article, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results. The notion of localization and related approximation properties introduced in this article are a spectrum of ideas that quantify, the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of the interrelations among these localization and approximation concepts is presented.

关键词： density excess frames Gabor systems modulation spaces overcompleteness Riesz bases wavelets Weyl-Heisenberg systems

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