Convolution Inequalities in Locally Compact Groups and Unitary Systems.
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| Title: | Convolution Inequalities in Locally Compact Groups and Unitary Systems. |
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| 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] |
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| Database: | Engineering Source |
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| 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] |
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| ISSN: | 01630563 |
| DOI: | 10.1080/01630563.2012.682142 |
