Abstract
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.
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1274–1279, September, 1995.
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Timan, M.F. On uniform approximations of almost periodic functions by entire functions of finite degree. Ukr Math J 47, 1449–1454 (1995). https://doi.org/10.1007/BF01057520
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DOI: https://doi.org/10.1007/BF01057520