On the relation between Fourier and Leont’ev coefficients with respect to smirnov spaces

Yu. Mel’nik showed that the Leont’ev coefficients Κf(λ) in the Dirichlet series $${{2n} \mathord{\left/ {\vphantom {{2n} {\left( {n + 1} \right) < p < 2}}} \right. \kern-\nulldelimiterspace} {\left( {n + 1} \right) < p > 2}}$$ of a function f ∈Ep(D), 1 < p < ∞, are the Fourier coefficients of some function F ∈Lp, ([0, 2π]) and that the first modulus of continuity of F can be estimated by the first moduli and majorants in f. In the present paper, we extend his results to moduli of arbitrary order.