We consider the dilation property of the modulation spaces M-p,M-q. Let D-lambda : f (t) -> (lambda t) be the dilation operator, and we consider the behavior of the operator norm parallel to D-lambda parallel to M-...
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We consider the dilation property of the modulation spaces M-p,M-q. Let D-lambda : f (t) -> (lambda t) be the dilation operator, and we consider the behavior of the operator norm parallel to D-lambda parallel to M-p,M-q -> M-p,M-q with respect to lambda. Our result determines the best order for it, and as an application, we establish the optimality of the inclusion relation between the modulation spaces and Besov spaces, which was proved by Toft [J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus, 1, J. Funct. Anal. 207 (2004) 399-429]. (c) 2007 Elsevier Inc. All rights reserved.
We deal with kernel theorems for modulation spaces. We completely characterize the continuity of a linear operator on the modulation spaces Mp for every 1p, by the membership of its kernel in (mixed) modulation spaces...
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We deal with kernel theorems for modulation spaces. We completely characterize the continuity of a linear operator on the modulation spaces Mp for every 1p, by the membership of its kernel in (mixed) modulation spaces. Whereas Feichtinger's kernel theorem (which we recapture as a special case) is the modulation space counterpart of Schwartz' kernel theorem for tempered distributions, our results do not have a counterpart in distribution theory. This reveals the superiority, in some respects, of the modulation space formalism over distribution theory, as already emphasized in Feichtinger's manifesto for a post-modern harmonic analysis, tailored to the needs of mathematical signal processing. The proof uses in an essential way a discretization of the problem by means of Gabor frames. We also show the equivalence of the operator norm and the modulation space norm of the corresponding kernel. For operators acting on Mp,q a similar characterization is not expected, but sufficient conditions for boundedness can be stated in the same spirit.
This paper provides sufficient conditions for the boundedness of Weyl operators on modulation spaces. The Weyl symbols belong to Wiener amalgam spaces, or generalized modulation spaces, as recently renamed by their in...
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This paper provides sufficient conditions for the boundedness of Weyl operators on modulation spaces. The Weyl symbols belong to Wiener amalgam spaces, or generalized modulation spaces, as recently renamed by their inventor Hans Feichtinger. This is the first result which relates symbols in Wiener amalgam spaces to operators acting on classical modulation spaces.
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 trans...
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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.
We give a new class of equivalent norms for modulation spaces by replacing the window of the short-time Fourier transform by a Hilbert-Schmidt operator. The main result is applied to Cohen's class of time-frequenc...
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We give a new class of equivalent norms for modulation spaces by replacing the window of the short-time Fourier transform by a Hilbert-Schmidt operator. The main result is applied to Cohen's class of time-frequency distributions, Weyl operators and localization operators. In particular, any positive Cohen's class distribution with Schwartz kernel can be used to give an equivalent norm for modulation spaces. We also obtain a description of modulation spaces as time-frequency Wiener amalgam spaces. The Hilbert-Schmidt operator must satisfy a nuclearity condition for these results to hold, and we investigate this condition in detail.
The alpha-modulation spaces M-p,q(s,alpha) (R-d) x is an element of [0, 1], form a family of spaces that include the Besov and modulation spaces as special cases. This paper is concerned with construction of Banach fr...
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The alpha-modulation spaces M-p,q(s,alpha) (R-d) x is an element of [0, 1], form a family of spaces that include the Besov and modulation spaces as special cases. This paper is concerned with construction of Banach frames for alpha-modulation spaces in the multivariate setting. The frames constructed are unions of independent Riesz sequences based on tensor products of univariate brushlet functions, which simplifies the analysis of the full frame. We show that the multivariate alpha-modulation spaces can be completely characterized by the Banach frames constructed. (c) 2005 Elsevier Inc. All rights reserved.
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 ...
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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 non-smooth symbols in the Feichtinger algebra are bounded on products of modulation spaces.
modulation spaces have received considerable interest recently as it is the natural function spaces to consider low regularity Cauchy data for several nonlinear evolution equations. We establish global well-posedness ...
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modulation spaces have received considerable interest recently as it is the natural function spaces to consider low regularity Cauchy data for several nonlinear evolution equations. We establish global well-posedness for 3D Klein-Gordon-Hartree equation u(tt) - Delta u + u + (vertical bar center dot vertical bar(-gamma) * vertical bar u vertical bar(2))u = 0 with initial data in modulation spaces M-1(p,p ') x M-p,M-p ' for p is an element of (2, 54 /27-2 gamma), 2 < gamma < 3. We implement Bourgain's high-low frequency decomposition method to establish global well-posedness, which was earlier used for classical Klein-Gordon equation. This is the first result on low regularity for Klein-Gordon-Hartree equation with large initial data in modulation spaces (which do not coincide with Sobolev spaces). (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The Cauchy problems for Navier-Stokes equations and nonlinear heat equations are studied in modulation spaces M(q,sigma)(s) (R(n)). Though the case of the derivative index s = 0 has been treated in our previous work. ...
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The Cauchy problems for Navier-Stokes equations and nonlinear heat equations are studied in modulation spaces M(q,sigma)(s) (R(n)). Though the case of the derivative index s = 0 has been treated in our previous work. the case s not equal 0 is also treated in this paper. Our aim is to reveal the conditions of s, q and sigma of M(q,sigma)(s) (R(n)) for the existence of local and global solutions for initial data u(0) is an element of M(q,sigma)(s) (R(n)). (C) 2009 Elsevier Inc All rights reserved
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