On 𝐿¹-norms of meromorphic functions with fixed poles

Abstract

We study boundedness of the differentiation and embedding operators in the shift-coinvariant subspaces K B 1 {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 L^1 -norm. We answer negatively the question of K.M. Dyakonov about the necessity of the condition B ′ ∈ L ∞ ( R ) B’\in L^\infty (\mathbb {R}) for the boundedness of the differentiation on K B 1 {K_B^1} . Our main tool is a construction of an unconditional basis of rational fractions in K B 1 {K_B^1} .