  The celebrated sparse representation model has led to remarkable results in
various signal processing tasks in the last decade. However, despite its
initial purpose of serving as a global prior for entire signals, it has been
commonly used for modeling low dimensional patches due to the computational
constraints it entails when deployed with learned dictionaries. A way around
this problem has been proposed recently, adopting a convolutional sparse
representation model. This approach assumes that the global dictionary is a
concatenation of banded Circulant matrices. Although several works have
presented algorithmic solutions to the global pursuit problem under this new
model, very few truly-effective guarantees are known for the success of such
methods. In the first of this two-part work, we address the theoretical aspects
of the sparse convolutional model, providing the first meaningful answers to
corresponding questions of uniqueness of solutions and success of pursuit
algorithms. To this end, we generalize mathematical quantities, such as the
$\ell_0$ norm, the mutual coherence and the Spark, to their counterparts in the
convolutional setting, which intrinsically capture local measures of the global
model. In a companion paper, we extend the analysis to a noisy regime,
addressing the stability of the sparsest solutions and pursuit algorithms, and
demonstrate practical approaches for solving the global pursuit problem via
simple local processing.
