ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2011, Vol. 55, No. 12, pp. 52–57. c Allerton Press, Inc., 2011. Original Russian Text c M.F. Timan and Yu.Kh. Khasanov, 2011, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2011, No. 12, pp. 64–70. Approximations of Almost Periodic Functions by Entire Ones M. F. Timan1* and Yu. Kh. Khasanov2** 1Dnepropetrovsk State Agrarian University, ul. Voroshilova 25, Dnepropetrovsk, 049027 Ukraine 2Russian-Tajik Slavonic University, 30 ul. M. Tursunzoda 30, Dushanbe, 734025 Tajikistan Received November 25, 2010 Abstract—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 (−∞, ∞) the best uniform approximation of order σ of periodic functions there exists a trigonometric polynomial whose order does not exceed σ. 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. DOI: 10.3103/S1066369X11120085 Keywords and phrases: almost periodic function, trigonometric polynomial, Fourier factors, uniform approximation, entire function of finite order, modulus of continuity. We denote by B(σ) the class of entire functions of order ≤ σ which are bounded on the whole real axis. S. N. Bernstein [1] has proved that among functions from the class B(σ), which realize the best uniform approximation of order σ of a periodic (of period 2π)functionf(x) on (−∞, ∞), there exists a trigonometric polynomial of degree not greater than σ. In this paper we give a new proof of the S. N. Bernstein theorem, which allows one to obtain the following results for uniform almost periodic functions. Definition. Afunctionf(x) is usually said to be uniform almost periodic, if for each ε>0 one can indicate a positive number l = l(ε) such that each interval of length l contains at least one number τ,for which |f(x + τ) − f(x)| <ε (−∞ <x<∞). For a uniform almost periodic function f(x), a set of real numbers Λ{λk} (k =1, 2,...)suchthat 1 T lim f(x)exp(−iλ x)dx =0 →∞ k T 2T −T is usually said to be the spectrum of f. Theorem 1. Let f(x) be a uniform almost periodic function with the spectrum Λ{λk} and let A(σ; f)(σ>0) be the best uniform approximation of order σ of the function f(x) from the class B(σ).Thenforanyε>0 there exists a finite trigonometric sum N P (x; N,σ)= bk exp(iλkx), (1) k=1 where λk ∈ Λ and |λk|≤σ (k =1, 2,...,N), such that |f(x) − P (x; N,σ)|≤A(σ; f)+ε (2) uniformly with respect to x. *E-mail: [email protected]. **E-mail: [email protected]. 52.