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For a fixed $\mathbf{h}$ in a subset of $\mathbb{C}^m$ such that $\mathbf{h}(k)\neq 0$ for any $k=0,...,m-1$, how can I find

$\sup_{\mathbf{x}} \{ \| \mathbf{x} \|_1 \,\,\, \mathrm{ s.t. } \,\,\, \|\mathbf{h}\ast\mathbf{x}\|_1 \leq 1 \} $,

where $\mathbf{x} \in \{ \mathbf{y} \in \mathbb{C}^n : \mathbf{h} \ast \mathbf{y} = 0 \Leftrightarrow \mathbf{y} = 0 \} \subset \mathbb{C}^n$ is a vector subspace and $\ast$ denotes convolution? For this, convolution between two vectors $\mathbf{u}\in\mathbb{C}^p$ and $\mathbf{v}\in\mathbb{C}^q$ is the full discrete convolution of length $p+q-1$:

$\mathbf{z}(k) = \sum_{j=-\infty}^{\infty} {\mathbf{u}(j) \mathbf{v}(k-j)}$,

and for the purposes of computation $\mathbf{u}(j)=0 \; \mathrm{for} \; j\notin[0,p-1]$ and $\mathbf{v}(j)=0 \; \mathrm{for} \; j\notin[0,q-1]$

I would like to understand how to characterize the above for any and all $\mathbf{h}$, however, I'd be happy to start with special cases. For instance, if $\mathbf{h}=[-1,+1]$, then $\|\mathbf{h}\ast\mathbf{x}\|_1$ is the discrete Total Variation, which is well-studied in the literature, albeit from a functional analysis approach.

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    @Norbert Than$k$s. I have added such a restrictio$n$ to the questio$n$.2012-06-12

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