On basicity of exponential and trigonometric systems in grand Lebesgue spaces

Abstract

Basis properties of exponential and trigonometric systems in grand Lebesgue spaces $ L_{p)} (-\pi,\pi) $ are studied. Based on a shift operator, we consider the subspace $G_{p)} (-\pi,\pi)$ of the space $ L_{p)} (-\pi,\pi) $, where continuous functions are dense, and the boundedness of the singular operator in this subspace is proved. We establish the basicity of exponential system $ \{{e^{int}}\}_{n\in Z}$ for $G_{p)} (-\pi,\pi)$ and the basicity of trigonometric systems $ \{{\sin{nt}}\}_{n\in N }$ and $ \{{\cos{nt}}\}_{n\in N_0}$ for $G_{p)} (0,\pi)$.