|
The approximation of piecewise smooth functions by trigonometric Fourier sums
M. G. Magomed-Kasumovab a Dagestan Federal research center of the RAS
b Vladikavkaz Scientific Centre of the Russian Academy of Sciences
Abstract:
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.
Received: 22.08.2019 Revised: 27.11.2019 Accepted: 28.11.2019
Citation:
M. G. Magomed-Kasumov, “The approximation of piecewise smooth functions by trigonometric Fourier sums”, Daghestan Electronic Mathematical Reports, 2019, no. 12, 25–42
Linking options:
https://www.mathnet.ru/eng/demr75 https://www.mathnet.ru/eng/demr/y2019/i12/p25
|
Statistics & downloads: |
Abstract page: | 189 | Full-text PDF : | 79 | References: | 42 |
|