Best approximations and widths of some classes of convolutions in $L_{2}$

We find tight upper bounds for the best approximations by trigonometric polynomials of certain classes of periodic functions representable as convolutions with structural characteristics defined by various modifications of m-th order moduli of continuity in the metric of $L_{2}$. We also find exact values for the $n$-widths of convolution classes given by such smoothness characteristics.