Approximations of almost periodic functions by entire ones

In this paper we propose a new proof of the well-known theorem by S. N. Bernstein, according to which among entire functions which give on $(-\infty,\infty)$ the best uniform approximation of order $\sigma$ of periodic functions there exists a trigonometric polynomial whose order does not exceed $\sigma$. We also prove an analog of this Bernstein theorem and an analog of the Jackson theorem for uniform almost periodic functions with an arbitrary spectrum.