On the equivalent bases of cosines in generalized Lebesgue spaces

In this paper the perturbed system of cosines is considered. The concept of ${\mathscr K}^{*} $-close systems generated by a space of coefficients ${\mathscr K}$ is introduced. Under certain conditions on the considered systems and on the space ${\mathscr K}$ the equivalence of bases of these systems in Lebesgue spaces with a variable summability exponent is established. This result generalizes the previous results in this direction.