Asymptotics of approximation of continuous periodic functions by linear means of their Fourier series

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.