Resolution limits for atomic decompositions via Markov-Bernstein type inequalities

Atomic norm offers a universal way to regularize ill-posed inverse problems. A fundamental problem in understanding the power of such regularization is determining norm-achieving decompositions. Sufficient conditions for atomic decomposition have been established for many instances of atomic norms. In this work, by using Markov-Bernstein type inequalities, we show that the resolution conditions that appear in these sufficient conditions are almost necessary.