Geodesic
Approximations of almost periodic functions by entire ones
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2011), pp. 64-70

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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.
Keywords: almost periodic function, trigonometric polynomial, Fourier factors, uniform approximation, entire function of finite order, modulus of continuity. 
@article{IVM_2011_12_a7,
     author = {M. F. Timan and Yu. Kh. Khasanov},
     title = {Approximations of almost periodic functions by entire ones},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {64--70},
     publisher = {mathdoc},
     number = {12},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2011_12_a7/}
}
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M. F. Timan; Yu. Kh. Khasanov. Approximations of almost periodic functions by entire ones. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2011), pp. 64-70. http://geodesic.mathdoc.fr/item/IVM_2011_12_a7/