  In this short article we present the theory of sparse representations
recovery in convex regularized optimization problems introduced in (Carioni and
Del Grande, arXiv:2311.08072, 2023). We focus on the scenario where the
unknowns belong to Banach spaces and measurements are taken in Hilbert spaces,
exploring the properties of minimizers of optimization problems in such
settings. Specifically, we analyze a Tikhonov-regularized convex optimization
problem, where $y_0$ are the measured data, $w$ denotes the noise, and
$\lambda$ is the regularization parameter. By introducing a Metric
Non-Degenerate Source Condition (MNDSC) and considering sufficiently small
$\lambda$ and $w$, we establish Exact Sparse Representation Recovery (ESRR) for
our problems, meaning that the minimizer is unique and precisely recovers the
sparse representation of the original data. We then emphasize the practical
implications of this theoretical result through two novel applications: signal
demixing and super-resolution with Group BLASSO. These applications underscore
the broad applicability and significance of our result, showcasing its
potential across different domains.
