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On the accuracy of the approximation of the complex exponent by the first terms of its Taylor expansion with applications. (English) Zbl 1311.41021

Summary: A modification of the Taylor expansion for the complex exponential function \(e^{ix}\), \(x\in\mathbb{R}\), is proposed yielding precise moment-type estimates of the accuracy of the approximation of a Fourier transform by the first terms of its Taylor expansion. Moreover, a precise upper bound for the third moment of a probability distribution in terms of the absolute third moment is established. Based on these results, new precise bounds for Fourier-Stieltjes transforms of probability distribution functions and for their derivatives are obtained that are uniform in classes of distributions with prescribed first three moments.

MSC:

41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
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