The best approximation to a class of functions of several variables by another class and related extremum problems

We study the relationship between several extremum problems for unbounded linear operators of convolution type in the spaces $L_\gamma=L_\gamma(\mathbb R^m)$, $m\ge1$, $1\le\gamma\le\infty$. For the problem of calculating the modulus of continuity of the convolution operator $A$ on the function class $Q$ defined by a similar operator and for the Stechkin problem on the best approximation of the operator $A$ on the class $Q$ by bounded linear operators, we construct dual problems in dual spaces, which are the problems on, respectively, the best and the worst approximation to a class of functions by another class.