Uniform approximations by Fourier sums on classes of convolutions of periodic functions

We establish asymptotic estimates for exact upper bounds of uniform approximations by Fourier sums on the classes of $2\pi$-periodic functions, which are represented by convolutions of functions $\phi (\phi \mathrm{\perp }1)$ from unit ball of the space $L_1$ with fixed kernels ${\mathrm{\Psi }}_{\beta }$ of the form $\Psi_{\beta}(t)=\sum\limits_{k=1}^{\infty}\psi(k) \cos\left(kt-\frac{\beta\pi}{2}\right)$,$\sum\limits_{k=1}^{\infty}k\psi(k)<\infty$,$\psi(k)\geq 0$,$\beta\in$\mathbb{R}$$.