In this paper the perturbed system of cosines is considered. Under certain conditions on the summability index p(·) and perturbation, the basicity of this system in Lebesgue spaces L p(·) (0, π) with variable summability index p(·) is proved. The obtained results generalize similar results for the case p(·) = p = const.

This paper deals with the symmetric space of functions and its subspace where continuous functions are dense is considered. Main properties of convolution which plays a vital role in harmonic analysis, as in other areas of mathematics are established in this space. Following the classical case, it is proved that the convolution can be approximated by linear combinations of shifts in a subspace of the considered space. An approximate identity for the convolution is also considered in that subspace.