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

M. I. Ismailov ; I. F. Aliyarova

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2024), pp. 15-25

Résumé

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 $G_{p),\rho } (0,\pi )$.