[Author]
 A. D. Baranov

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

[AMS Subj-class]
 30D50 Blaschke products, bounded mean oscillation, bounded characteristic,
       bounded functions, functions with positive real part
 30D55 $H^p$-classes
 46E15 Banach spaces of continuous, differentiable or analytic functions
 47B38 Operators on function spaces (general)

[Abstract]
 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 $K_B^1$.

[Keywords]
 Blaschke products, shift-coinvariant subspaces, Bernstein's inequality,
 unconditional basis

[Comments]
 LaTeX, English, 15 pp.

[Contact e-mail]
 antonbaranov@netscape.net
 d.baranov@pop.ioffe.rssi.ru