Abstract
In the present article, the approximation of the function |sin x| s by the partial sums of the rational trigonometric Fourier series is considered. An integral representation, uniform and point estimates for the above-mentioned approximation were obtained. Based on them, several special cases of the selection of poles were studied. In the case of the approximation by the partial sums of the polynomial trigonometric Fourier series, an asymptotic equality was found. A detailed study is made of a fixed number of geometrically different poles.
Highlights
In the present article, the approximation of the function |sin x|s by the partial sums of the rational trigonometric Fourier series is considered
A detailed study is made of a fixed number of geometrically different poles
Для набліжэнняў функцыі |sin x|s, s > 0, частковымі сумамі трыганаметрычнага рацыянальнага шэрагу Фур’е праўдзіцца няроўнасць ε 2n sin πs 2 ne −π ns
Summary
The approximation of the function |sin x|s by the partial sums of the rational trigonometric Fourier series is considered. Для функцыі f(x) = |sin x|s, s > 0, x R, разгледзім частковыя сумы яе трыганаметрычнага рацыянальнага шэрагу Фур’е [6] Для набліжэння функцыі |sin x|s частковымі сумамі трыганаметрычных рацыянальных шэрагаў Фур’е праўдзіцца наступная роўнасць: ε 2n (x, a)
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