zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Annihilating sets for the short time Fourier transform. (English) Zbl 1194.42015
The uncertainty principle means that a non-zero function and its Fourier transform cannot be sharply localized. In this paper, the authors consider support conditions for the short time Fourier transform (STFT). The aim is to obtain a class of subsets in ${\mathbb R}^{2d}$ (called thin sets at infinity) so that the support of the STFT of a signal $f\in L^2({\mathbb R}^d)$ with respect to a non-zero window $g\in L^2({\mathbb R}^d)$ cannot belong to this class unless $f=0$. Moreover it is proved that the $L^2$-norm of the STFT is essentially concentrated in the complement of any thin set at infinity. Finally, this result is generalized to other Hilbert spaces of functions or distributions.

MSC:
42B10Fourier type transforms, several variables
WorldCat.org
Full Text: DOI
References:
[1] Benedicks, M.: On Fourier transforms of functions supported on sets of finite Lebesgue measure, J. math. Anal. appl. 106, 180-183 (1985) · Zbl 0576.42016 · doi:10.1016/0022-247X(85)90140-4
[2] Boggiatto, P.: Localization operators with lp symbols on modulation spaces, Advances in pseudodifferential operators, operator theory: advances and appl., proc. Isaac congress (2003)
[3] Boggiatto, P.; Cordero, E.; Gröchenig, K.: Generalized anti-Wick operators with symbols in distributional Sobolev spaces, Integral equations operator theory 48, 427-442 (2004) · Zbl 1072.47045 · doi:10.1007/s00020-003-1244-x
[4] Boggiatto, P.; De Donno, G.; Oliaro, A.: Uncertainty principle, positivity and lp-boundedness for generalized spectrograms, J. math. Anal. appl. 335, 93-112 (2007) · Zbl 1143.42013 · doi:10.1016/j.jmaa.2007.01.019
[5] Bonami, A.; Demange, B.: A survey on uncertainty principles related to quadratic forms, Collect. math., 1-36 (2006) · Zbl 1107.30021
[6] Cordero, E.; Gröchenig, K.: Time-frequency analysis of localization operators, J. funct. Anal. 205, 107-131 (2003) · Zbl 1047.47038 · doi:10.1016/S0022-1236(03)00166-6
[7] Daubechies, I.: Time-frequency localization operators: a geometric phase space approach, IEEE trans. Inform. theory 34, 605-612 (1988) · Zbl 0672.42007 · doi:10.1109/18.9761
[8] Demange, B.: Uncertainty principles for the ambiguity function, J. London math. Soc. 72, No. 2, 717-730 (2005) · Zbl 1090.42004 · doi:10.1112/S0024610705006903
[9] Fernández, C.; Galbis, A.: Compactness of time-frequency localization operators on $L2(Rd)$, J. funct. Anal. 233, 335-350 (2006) · Zbl 1100.47039 · doi:10.1016/j.jfa.2005.08.008
[10] Fernández, C.; Galbis, A.: Some remarks on compact Weyl operators, Integral transforms spec. Funct. 18, 599-607 (2006) · Zbl 1131.47029 · doi:10.1080/10652460701445476
[11] Folland, G.; Sitaram, A.: The uncertainty principle: a mathematical survey, J. Fourier anal. Appl. 3, 207-238 (1997) · Zbl 0885.42006 · doi:10.1007/BF02649110
[12] Gröchenig, K.: Foundations of time-frequency analysis, (2001) · Zbl 0966.42020
[13] Gröchenig, K.: Uncertainty principles for time-frequency representations, Appl. numer. Harmon. anal., 11-30 (2003) · Zbl 1039.42004
[14] Havin, V.; Jöricke, B.: The uncertainty principle in harmonic analysis, Encyclopaedia math. Sci. 72 (1995) · Zbl 0826.42001
[15] Jaming, Ph.: Principe d’incertitude qualitatif et reconstruction de phase pour la transformée de Wigner, C. R. Acad. sci. Ser. I math. 237, 249-254 (1998) · Zbl 0931.42006 · doi:10.1016/S0764-4442(98)80141-9
[16] Jaming, Ph.: Nazarov’s uncertainty principle in higher dimension, J. approx. Theory 149, 611-630 (2007) · Zbl 1119.42012
[17] Janssen, A. J. E.M.: Proof of a conjecture on the supports of Wigner distributions, J. Fourier anal. Appl. 4, 723-726 (1998) · Zbl 0924.42009 · doi:10.1007/BF02479675
[18] Shubin, C.; Vakilian, R.; Wolff, T.: Some harmonic analysis questions suggested by Anderson-Bernoulli models, Geom. funct. Anal. 8, 932-964 (1998) · Zbl 0920.42005 · doi:10.1007/s000390050078
[19] Toft, J.: Continuity and Schatten properties for Toeplitz operators on modulation spaces, Oper. theory adv. Appl. 172, 173-206 (2007) · Zbl 1133.35110 · doi:10.1007/978-3-7643-8116-5_11
[20] Toft, J.: Continuity and Schatten properties for pseudo-differential operators on modulation spaces, Oper. theory adv. Appl. 172, 173-206 (2007) · Zbl 1133.35110 · doi:10.1007/978-3-7643-8116-5_11
[21] Toft, J.; Boggiatto, P.: Schatten classes for Toeplitz operators with Hilbert space windows on modulation spaces, Adv. math. 217, 305-333 (2008) · Zbl 1131.47026 · doi:10.1016/j.aim.2007.07.001
[22] Wilczok, E.: New uncertainty principles for the Gabor transform and the continuous wavelet transform, Doc. math. 5, 201-226 (2000) · Zbl 0947.42024 · emis:journals/DMJDMV/vol-05/08.html