On uniform approximations of almost periodic functions by entire functions of finite degree - Timan, M. F.; Тиман, М. Ф.; Тиман, М. Ф. (Institute of Mathematics, NAS of Ukraine, 1995)
We give a new proof of the well-known Bernshtein statement that, among entire functions of degree $≤ σ$ which realize the best uniform approximation (of degree $σ$) of a periodic function on $(−∞,∞)$, there is a trigonometric polynomial of degree $≤ σ$. We prove an analog of the mentioned Bernshtein statement and the Jackson theorem for uniform almost periodic functions with arbitrary spectrum.