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- Title
Continuous Partial Gabor Transform for Semi-Direct Product of Locally Compact Groups.
- Authors
Ghaani Farashahi, Arash
- Abstract
Let $$H$$ be a locally compact group, $$K$$ be an LCA group, $$\tau :H\rightarrow Aut(K)$$ be a continuous homomorphism and $$G_\tau =H\ltimes _\tau K$$ be the semi-direct product of $$H$$ and $$K$$ with respect to the continuous homomorphism $$\tau $$ . In this article, we introduce the $$\tau \times \widehat{\tau }$$ -time frequency group $$G_{\tau \times \widehat{\tau }}$$ . We define the $$\tau \times \widehat{\tau }$$ -continuous Gabor transform of $$f\in L^2(G_\tau )$$ with respect to a window function $$u\in L^2(K)$$ as a function defined on $$G_{\tau \times \widehat{\tau }}$$ . It is also shown that the $$\tau \times \widehat{\tau }$$ -continuous Gabor transform satisfies the Plancherel Theorem and reconstruction formula. This approach is tailored for choosing elements of $$L^2(G_\tau )$$ as a window function. Finally, we indicate some possible applications of these methods in the case of some well-known semi-direct product groups.
- Subjects
GABOR transforms; HOMOMORPHISMS; COMPACT groups; TIME-frequency analysis; PARTIAL algebras
- Publication
Bulletin of the Malaysian Mathematical Sciences Society, 2015, Vol 38, Issue 2, p779
- ISSN
0126-6705
- Publication type
Article
- DOI
10.1007/s40840-014-0049-1