Cosine and sine series with generalized monotone coefficients in Lorentz spaces

We obtain a necessary and sufficient condition for the sums f or g of cosine or sine series with general monotone coefficients to belong to the Lorentz space I >(q, I center dot) in terms of these coefficients {a (n) } (n=1) (a) . Similar results are also obtained for weighted L (q) spaces. As a corollary, the equivalence of norms of f or g in Lorentz space and corresponding weighted L (q) space with norms of {a (n) } (n=1) (a) in appropriate weighted l (q) space is established. The results of this paper generalize earlier results of Booton, Dyachenko, Simonov and Tikhonov.