Geodesic
The approximation of piecewise smooth functions by trigonometric Fourier sums
Daghestan Electronic Mathematical Reports, Tome 12 (2019), pp. 25-42.

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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, convergence rate, piecewise linear functions. 
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M. G. Magomed-Kasumov. The approximation of piecewise smooth functions by trigonometric Fourier sums. Daghestan Electronic Mathematical Reports, Tome 12 (2019), pp. 25-42. http://geodesic.mathdoc.fr/item/DEMR_2019_12_a2/

[1] Kolmogoroff A. N., “Zur Grossenordnung Des Restgliedes Fourierscher Reihen Differenzierbarer Funktionen”, Annals of Mathematics, 36:2 (1935), 521–526 | DOI

[2] Magomed-Kasumov M.G., “Approksimativnye svoistva klassicheskikh srednikh Valle-Pussena dlya kusochno gladkikh funktsii”, Vestnik Dagestanskogo nauchnogo tsentra RAN, 54 (2014), 5–12

[3] Magomed-Kasumov M.G., “Approksimativnye svoistva srednikh Valle Pussena dlya kusochno gladkikh funktsii”, Matematicheskie zametki, 100:2 (2016), 229–247 | Zbl

[4] Sharapudinov I.I., Magomed-Kasumov M.G. , Sharapudinov T.I., “Approksimativnye svoistva povtornykh srednikh Valle Pussena dlya kusochno-gladkikh funktsii”, Sibirskii matematicheskii zhurnal (otpravlena v pechat)

[5] Fikhtengolts G.M., Kurs differentsialnogo i integralnogo ischisleniya, FIZMATLIT, Moskva, 2001, 810 pp.

[6] Zigmund A., Trigonometricheskie ryady, v. 1, Mir, Moskva, 1965, 615 pp.

[7] Magomed-Kasumov M.G., “Otsenka skorosti skhodimosti ryadov po sinusam i kosinusam s koeffitsientami vida $1/k^q$”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 7, 47–51

[8] Akniev G.G., “Approksimativnye svoistva summ Fure dlya $2\pi$-periodicheskikh kusochno-lineinykh nepreryvnykh funktsii”, Dagestanskie elektronnye matematicheskie izvestiya, 2016, no. 5, 13–19