Geodesic
The approximation of piecewise smooth functions by trigonometric Fourier sums
Daghestan Electronic Mathematical Reports, no. 12 (2019), pp. 25-42 Cet article a éte moissonné depuis la source Math-Net.Ru

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We obtain exact order-of-magnitude estimates of piecewise smooth functions approximation by trigonometric Fourier sums. It is shown that in continuity points Fourier series of piecewise Lipschitz function converges with rate $\ln n/n$. If function $f$ has a piecewise absolutely continuous derivative then it is proven that in continuity points decay order of Fourier series remainder $R_n(f,x)$ for such function is equal to $1/n$. We also obtain exact order-of-magnitude estimates for $q$-times differentiable functions with piecewise smooth $q$-th derivative. In particular, if $f^{(q)}(x)$ is piecewise Lipschitz then $|R_n(f,x)| \le c(x)\frac{\ln n}{n^{q+1}}$ in continuity points of $f^{(q)}(x)$ and $\sup_{x \in [0,2\pi]}|R_n(f,x)| \le \frac{c}{n^q}$. In case when $f^{(q)}(x)$ has piecewise absolutely continuous derivative it is shown that $|R_n(f,x)| \le \frac{c(x)}{n^{q+1}}$ in continuity points of $f^{(q)}(x)$. As a consequence of the last result convergence rate estimate of Fourier series to continuous piecewise linear functions is obtained.
Keywords: piecewise smooth functions, Fourier series, piecewise linear functions.
Mots-clés : convergence rate 
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     title = {The approximation of piecewise smooth functions by trigonometric {Fourier} sums},
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M. G. Magomed-Kasumov. The approximation of piecewise smooth functions by trigonometric Fourier sums. Daghestan Electronic Mathematical Reports, no. 12 (2019), pp. 25-42. http://geodesic.mathdoc.fr/item/DEMR_2019_12_a2/

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