The Hardy-Littlewood theorem for the cosine series in a symmetric space

For a wide class of functional spaces we obtain a necessary and sufficient condition on a space that guarantees a Hardy–Littlewood type of assertion about whether the sum of a cosine series with monotonic coefficients belongs to a functional space, e.g., $L_p$ ($p>1$). As examples we consider Lorentz spaces, Marcinkiewicz spaces, Orlicz spaces, and $L_p$ spaces.