On $L^1$-norms of meromorphic functions with fixed poles

We study boundedness of the differentiation and embedding operators in the shift-coinvariant subspaces ${K_B^1}$ generated by Blaschke products with sparse zeros, that is, in the spaces of meromorphic functions with fixed poles in the lower half-plane endowed with $L^1$-norm. We answer negatively the question of K.M. Dyakonov about the necessity of the condition $B'\in L^\infty(\mathbb{R})$ for the boundedness of the differentiation on ${K_B^1}$. Our main tool is a construction of an unconditional basis of rational fractions in ${K_B^1}$.