On Best Polynomial Approximations in $L_2$

For the $\boldsymbol\tau$-moduli of smoothness of $m$th order, we calculate exact constants in Jackson-type inequalities. We also obtain the exact values of the $n$-widths of classes of functions whose $r$th derivatives are characterized by $\boldsymbol\tau$-moduli of smoothness majorized by functions satisfying certain constraints. We present an example of the majorant for which all the stated requirements are satisfied.