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Convolution Inequalities in Locally Compact Groups and Unitary Systems.

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Title: Convolution Inequalities in Locally Compact Groups and Unitary Systems.
Authors: Gabardo, Jean-Pierre1 (AUTHOR) gabardo@mcmaster.ca
Source: Numerical Functional Analysis & Optimization. Aug2012, Vol. 33 Issue 7-9, p1005-1030. 26p.
Subjects: Mathematical inequalities, Locally compact groups, Vector analysis, Radon measures, Hilbert space, Wavelets (Mathematics), Mathematical convolutions
Abstract: We consider certain convolution inequalities for positive Radon measures on a locally compact group G, also assumed σ-compact. These appear naturally in connection with Bessel or frame inequalities for certain unitary systems U t , t, ∈ G, of operators acting on a Hilbert space ℋ and associated with a positive Radon measure μ on G and an analyzing vector ψ ∈ ℋ. Using this approach, we obtain some general results in the form of inequalities relating the Bessel or frame constants to other constants defined in terms of the measure μ and the analyzing vector ψ. Specific examples are considered in the case where G is the Euclidean space ℝ d for the windowed exponentials and Gabor systems as well as in the case where G is the ax + b group for the wavelet systems. In particular, we construct examples of non-admissible wavelets generating (irregular) Parseval frame wavelet systems for L 2(ℝ). [ABSTRACT FROM AUTHOR]
Copyright of Numerical Functional Analysis & Optimization is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="DE" term="%22Mathematical+inequalities%22">Mathematical inequalities</searchLink><br /><searchLink fieldCode="DE" term="%22Locally+compact+groups%22">Locally compact groups</searchLink><br /><searchLink fieldCode="DE" term="%22Vector+analysis%22">Vector analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Radon+measures%22">Radon measures</searchLink><br /><searchLink fieldCode="DE" term="%22Hilbert+space%22">Hilbert space</searchLink><br /><searchLink fieldCode="DE" term="%22Wavelets+%28Mathematics%29%22">Wavelets (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+convolutions%22">Mathematical convolutions</searchLink>
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  Data: We consider certain convolution inequalities for positive Radon measures on a locally compact group G, also assumed σ-compact. These appear naturally in connection with Bessel or frame inequalities for certain unitary systems U t , t, ∈ G, of operators acting on a Hilbert space ℋ and associated with a positive Radon measure μ on G and an analyzing vector ψ ∈ ℋ. Using this approach, we obtain some general results in the form of inequalities relating the Bessel or frame constants to other constants defined in terms of the measure μ and the analyzing vector ψ. Specific examples are considered in the case where G is the Euclidean space ℝ d for the windowed exponentials and Gabor systems as well as in the case where G is the ax + b group for the wavelet systems. In particular, we construct examples of non-admissible wavelets generating (irregular) Parseval frame wavelet systems for L 2(ℝ). [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Numerical Functional Analysis & Optimization is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1080/01630563.2012.682142
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      – Code: eng
        Text: English
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        PageCount: 26
        StartPage: 1005
    Subjects:
      – SubjectFull: Mathematical inequalities
        Type: general
      – SubjectFull: Locally compact groups
        Type: general
      – SubjectFull: Vector analysis
        Type: general
      – SubjectFull: Radon measures
        Type: general
      – SubjectFull: Hilbert space
        Type: general
      – SubjectFull: Wavelets (Mathematics)
        Type: general
      – SubjectFull: Mathematical convolutions
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      – TitleFull: Convolution Inequalities in Locally Compact Groups and Unitary Systems.
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              Text: Aug2012
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