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97
Hardy's Theorem And The ShortTime Fourier Transform Of Schwartz Functions
, 2001
"... We characterize the Schwartz space of rapidly decaying test functions by the decay of the shorttime Fourier transform or crossWigner distribution. Then we prove a version of Hardy's theorem for the shorttime Fourier transform and for the Wigner distribution. ..."
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Cited by 36 (7 self)
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We characterize the Schwartz space of rapidly decaying test functions by the decay of the shorttime Fourier transform or crossWigner distribution. Then we prove a version of Hardy's theorem for the shorttime Fourier transform and for the Wigner distribution.
Hermite Functions And Uncertainty Principles For The Fourier And The Windowed Fourier Transforms
 Rev. Mat. Iberoamericana
"... We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on R which may be written as P (x) exp(Ax; x), with A a real symmetric de nite positive matrix, are characterized by integrability conditions on the product f(x) f(y) ..."
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Cited by 31 (2 self)
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We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on R which may be written as P (x) exp(Ax; x), with A a real symmetric de nite positive matrix, are characterized by integrability conditions on the product f(x) f(y). We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenberg's inequality for this transform. 1.
Resolution of singularities in DenjoyCarleman classes
 Selecta Math. (N.S
"... Abstract. We show that a version of the desingularization theorem of Hironaka holds for certain classes of C ∞ functions (essentially, for subrings that exclude flat functions and are closed under differentiation and the solution of implicit equations). Examples are quasianalytic classes, introduced ..."
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Cited by 26 (2 self)
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Abstract. We show that a version of the desingularization theorem of Hironaka holds for certain classes of C ∞ functions (essentially, for subrings that exclude flat functions and are closed under differentiation and the solution of implicit equations). Examples are quasianalytic classes, introduced by E. Borel a century ago and characterized by the DenjoyCarleman theorem. These classes have been poorly understood in dimension> 1. Resolution of singularities can be used to obtain many new results; for example, topological Noetherianity, ̷Lojasiewicz inequalities, division properties.
Uncertainty principles and vector quantization
"... An abstract form of the Uncertainty Principle set forth by Candes and Tao has found remarkable applications in the sparse approximation theory. This paper demonstates a new connection between the Uncertainty Principle and the vector quantization theory. We show that for frames in C n that satisfy ..."
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Cited by 20 (0 self)
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An abstract form of the Uncertainty Principle set forth by Candes and Tao has found remarkable applications in the sparse approximation theory. This paper demonstates a new connection between the Uncertainty Principle and the vector quantization theory. We show that for frames in C n that satisfy the Uncertainty Principle, one can quickly convert every frame representation into a more regular Kashin’s representation, whose coefficients all have the same magnitude O(1 /√n). Information tends to spread evenly among these coefficients. As a consequence, Kashin’s representations have a great power for reduction of errors in their coefficients. In particular, scalar quantization of Kashin’s representations yields robust vector quantizers in C n.
MEROMORPHIC INNER FUNCTIONS, TOEPLITZ KERNELS, AND THE UNCERTAINTY PRINCIPLE
"... This paper touches upon several traditional topics of 1D linear complex analysis including distribution of zeros of entire functions, completeness problem for complex exponentials and for other families of special functions, some problems of spectral theory of selfadjoint differential operators. The ..."
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Cited by 19 (3 self)
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This paper touches upon several traditional topics of 1D linear complex analysis including distribution of zeros of entire functions, completeness problem for complex exponentials and for other families of special functions, some problems of spectral theory of selfadjoint differential operators. Their common feature is the close relation
Uncertainty Principles in Hilbert Spaces
, 2002
"... In this paper we provide several generalizations of inequalities bounding the commutator of two linear operators acting on a Hilbert space which relate to the Heisenberg uncertainty principle and time/frequency analysis of periodic functions. We develop conditions that ensure these inequalities a ..."
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Cited by 14 (5 self)
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In this paper we provide several generalizations of inequalities bounding the commutator of two linear operators acting on a Hilbert space which relate to the Heisenberg uncertainty principle and time/frequency analysis of periodic functions. We develop conditions that ensure these inequalities are sharp and apply our results to concrete examples of importance in the literature.
Some results related to the LogvinenkoSereda theorem
 Proc. Am. Math. Soc
"... We prove several results related to the theorem of Logvinenko and Sereda on determining sets for functions with Fourier transforms supported in an interval. We obtain a polynomial instead of exponential bound in this theorem, and we extend it to the case of functions with Fourier transforms supporte ..."
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Cited by 13 (0 self)
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We prove several results related to the theorem of Logvinenko and Sereda on determining sets for functions with Fourier transforms supported in an interval. We obtain a polynomial instead of exponential bound in this theorem, and we extend it to the case of functions with Fourier transforms supported in the union of a bounded number of intervals. The same results hold in all dimensions. The purpose of this work is to study the behavior of functions whose Fourier transforms are supported in an interval or in a union of finitely many intervals on “thick ” subsets of the real line. A main result of this type was proven by Logvinenko and Sereda. By a “thick ” subset of the real line we mean a measurable set E for which there exist a> 0 and γ> 0 such that for every interval I of length a. E ∩ I  ≥ γ · a (1) The LogvinenkoSereda Theorem: let J be an interval with J  = b. If f ∈ L p, p ∈ [1, +∞], and supp ˆ f ⊂ J and if a measurable set E satisfies (1) then ‖f‖L p (E) ≥ exp(−C ·
A (p, q) version of Bourgain’s Theorem
 Trans. Amer. Math. Soc
"... Abstract. Let 1 < p, q < ∞ satisfy 1 1 ..."
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FOURIER TAUBERIAN THEOREMS AND APPLICATIONS
, 2000
"... Let F be a nondecreasing function and ρ is an appropriate test function on the real line R. Then, under certain conditions on the Fourier transform of the convolution ρ ∗ F, one can estimate the difference F − ρ ∗ F. Results of this type are called Fourier Tauberian theorems. The Fourier Tauberian ..."
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Cited by 8 (2 self)
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Let F be a nondecreasing function and ρ is an appropriate test function on the real line R. Then, under certain conditions on the Fourier transform of the convolution ρ ∗ F, one can estimate the difference F − ρ ∗ F. Results of this type are called Fourier Tauberian theorems. The Fourier Tauberian theorems have been used by many authors for the study of spectral asymptotics of elliptic differential operators, with F being either the counting function or the spectral function (see, for example, [L], [H1], [H2], [DG], [I1], [I2], [S], [SV]). The required estimate for F −ρ∗F was obtained under the assumption that the derivative ρ∗F ′ admits a sufficiently good estimate. In applications F often depends on additional parameters and we are interested in estimates which are uniform with respect to these parameters. Then one has to assume that the estimate for ρ ∗ F ′ holds uniformly and to take this into account when estimating F − ρ ∗ F. As a result, there have been