Geodesic
Asymptotics of~approximation of~continuous periodic functions by linear means of~their Fourier series
Izvestiya. Mathematics , Tome 84 (2020) no. 3, pp. 608-624

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We establish an asymptotic formula for the rate of approximation of Fourier series of individual periodic functions by linear averages with an error $\omega_{2m}(f;{1}/{n})$, $m\in\mathbb{N}$. This formula is applicable to the means of Riesz, Gauss–Weierstrass, Picard and others. The result is new even for the arithmetic means of partial Fourier sums. We use the formula to determine the asymptotic behaviour of functions in a certain class. Separately, we consider the case of positive integral convolution operators.
Keywords: Fourier series, Wiener algebra of Fourier transforms, comparison principle, modulus of smoothness $\omega_m(f;h)$, positive definite functions, Bernstein's and Schoenberg's theorems. 
@article{IM2_2020_84_3_a5,
     author = {R. M. Trigub},
     title = {Asymptotics of~approximation of~continuous periodic functions by linear means of~their {Fourier} series},
     journal = {Izvestiya. Mathematics },
     pages = {608--624},
     publisher = {mathdoc},
     volume = {84},
     number = {3},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2020_84_3_a5/}
}
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R. M. Trigub. Asymptotics of~approximation of~continuous periodic functions by linear means of~their Fourier series. Izvestiya. Mathematics , Tome 84 (2020) no. 3, pp. 608-624. http://geodesic.mathdoc.fr/item/IM2_2020_84_3_a5/