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 Λ(q, ϕ) in terms of these coefficients {an}n=1∞. Similar results are also obtained for weighted Lq spaces. As a corollary, the equivalence of norms of f or g in Lorentz space and corresponding weighted Lq space with norms of {an}n=1∞ in appropriate weighted lq space is established. The results of this paper generalize earlier results of Booton, Dyachenko, Simonov and Tikhonov.