Best Approximation and Symmetric Decreasing Rearrangements of Functions

The problem of estimating the best approximation by a subspace of classes of functions of $n$ variables defined by restrictions imposed on the modulus of continuity is considered on the basis of the duality principle. An approach is analyzed that is connected with the representation of a function of $n$ variables as a countable sum of simple functions and the subsequent transition to spatial symmetric decreasing rearrangements.