"Generalized Regular Sampling for Multiplication Invariant Subspaces"Sahil, SahilThe theory of sampling has widely been discussed for the shift invariant (SI) subspaces. The SI subspaces are closely related to multiplication invariant (MI) subspaces, which was first proposed by M. Bownik and K. Ross and then studied further by M. Bownik and J. W. Iverson. The relation between aforementioned types of invariant subspaces is facilitated by the Zak transform which converts translations in $L^2(G)$ into multiplications in $L^2(\hat{H}; L^2(H\backslash G))$ where $G$ is a locally compact group and $H$ is some abelian subgroup of $G$. In the present work, we consider $L^2(X;\mathcal{H})$, which consists of vector valued functions from a measure space $X$ to a separable Hilbert space $\mathcal{H}$. We obtain a sampling formula for the MI subspaces of $L^2(X;\mathcal{H})$ using the generalized samples, similar to the sampling formula obtained for the SI subspaces by H.R. Fernandez Morales. Motivated by the work of A. Aldroubi et al. , we also obtain some analogous conditions under which the set of sampling becomes stable. Along with this, we investigate the error in sampling by taking perturbed samples using an error sequence. We also provide two applications of our results. One, we produce a sampling formula for a locally compact group $G$ using the Zak transform. Another, we derive a sampling formula for the closed subspace of a separable Hilbert space generated by admissible unitary representations. |
| http://univie.ac.at/projektservice-mathematik/e/talks/ICCHA2022_Sahil_2022-05_abstract.pdf |
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