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- W2949467564 abstract "Let $G=NH$ be a Lie group where $N,H$ are closed connected subgroups of $G,$ and $N$ is an exponential solvable Lie group which is normal in $G.$ Suppose furthermore that $N$ admits a unitary character $chi_{lambda}$ corresponding to a linear functional $lambda$ of its Lie algebra. We assume that the map $hmapsto Adleft( h^{-1}right) ^{ast}lambda$ defines an immersion at the identity of $H$. Fixing a Haar measure on $H,$ we consider the unitary representation $pi$ of $G$ obtained by inducing $chi_{lambda}.$ This representation which is realized as acting in $L^{2}left( H,dmu_{H}right) $ is generally not irreducible, and we do not assume that it satisfies any integrability condition. One of our main results establishes the existence of a countable set $Gammasubset G$ and a function $mathbf{f}in L^{2}left( H,dmu_{H}right) $ which is compactly supported and bounded such that $left{ pileft( gammaright) mathbf{f}:gammainGammaright} $ is a frame. Additionally, we prove that $mathbf{f}$ can be constructed to be continuous. In fact, $mathbf{f}$ can be taken to be as smooth as desired. Our findings extend the work started in cite{oussa2018frames} to the more general case where $H$ is any connected Lie group. We also solve a problem left open in cite{oussa2018frames}. Precisely, we prove that in the case where $H$ is an exponential solvable group, there exist a continuous (or smooth) function $mathbf{f}$ and a countable set $Gamma$ such that $left{ pileft( gammaright) mathbf{f}:gammainGammaright} $ is a Parseval frame. Since the concept of well-localized frames is central to time-frequency analysis, wavelet, shearlet and generalized shearlet theories, our results are relevant to these topics and our approach leads to new constructions which bear potential for applications." @default.
- W2949467564 created "2019-06-27" @default.
- W2949467564 creator A5017710435 @default.
- W2949467564 date "2018-11-26" @default.
- W2949467564 modified "2023-09-27" @default.
- W2949467564 title "Compactly supported bounded frames on Lie groups" @default.
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