A Counterexample to Convergence for Multiscale Decompositions

We discuss the convergence of the multiscale procedure by Modin et al. (Adv. Math. 346:1009–1066, 2019), which extended to inverse problems the multiscale decomposition of images by Tadmor et al. (Multiscale Model. Simul. 2:554–579, 2004). We show that, for the classical multiscale procedure, the multiscale decomposition might fail even for the linear case with a Banach norm as regularization.