On the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in weighted grand Lebesgue spaces

The paper is focused on the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in a separable subspace of the weighted grand Lebesgue space generated by the shift operator. In this paper, with the help of the shift operator, a separable subspace $G_{p),\rho } (a,b)$ of the weighted space of the grand Lebesgue space $L_{p),\rho } (a,b)$ is defined. The density in $G_{p),\rho } (a,b)$ of the set $G_{0}^{\infty } ([a,b])$ of infinitely differentiable and finite on $[a,b]$ functions is studied. It is proved that if the weight function $\rho$ satisfies the Mackenhoupt condition, then the system of exponentials $\left\{e^{int} \right\}_{n\in Z}$ forms a basis in $G_{p),\rho } (-\pi ,\pi )$, and trigonometric systems of sine $\left\{\sin nt\right\}_{n\ge 1}$ and cosine $\left\{\cos nt\right\}_{n\ge 0}$ functions form bases in $G_{p),\rho } (0,\pi )$.