We establish an asymptotic formula for the rate of approximation of Fourier series of individual periodic functions by linear averages with an error ##IMG## [http://ej.iop.org/images/1064-5632/84/3/608/IZV_84_3_608ieqn1.gif] {$\omega_{2m}(f;{1}/{n})$} , ##IMG## [http://ej.iop.org/images/1064-5632/84/3/608/IZV_84_3_608ieqn2.gif] {$m\in\mathbb{N}$} . 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.

中文翻译：

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通过线性傅里叶级数逼近连续周期函数的渐近性

我们建立了线性平均的傅立叶级数与周期平均值的近似率的渐近公式，误差为## IMG ## [http://ej.iop.org/images/1064-5632/84/3/608 /IZV_84_3_608ieqn1.gif] {$ \ omega_ {2m}（f; {1} / {n}）$}，## IMG ## [http://ej.iop.org/images/1064-5632/84/ 3/608 / IZV_84_3_608ieqn2.gif] {$ m \ in \ mathbb {N} $}。此公式适用于Riesz，Gauss-Weierstrass，Picard和其他方法。即使对于部分傅立叶和的算术平均值，结果也是新的。我们使用公式来确定某个类中函数的渐近行为。另外，我们考虑正积分卷积算子的情况。