Exact values of relative widths of classes of differentiable functions (Q1974383)

Let \(MW_C^r\), \(r=1,2,\ldots\), be the class of \(2\pi\)-periodic functions \(f\) satisfying \[ |f ^{(r-1)} (x_1)-f^{(r-1)} (x_2)|\leq M|x_1-x_2|, \] and write \(W_C^r\) for \(M=1\). The paper studies the relative \(n\)-dimensional width \(K_n(W_C^r, MW_C^r, C)\) defined by \[ K_n(W_C^r, MW_C^r, C):=\inf_{L_n} \sup_{f\in W_C^r}\inf_{g\in L_n\cap MW_C^r}\|f-g\|_C, \] where \(L_n\) is a linear subspace of \(C\) of dimension \(n\). It is proved that if \(M\) satisfies the estimate \[ M\geq {4\over\pi^2} \log \min (n,r)+ O (1), \] then the relative \(n\)--dimensional widths coincide with the corresponding Kolmogorov widths. Analogous results are shown for the space \(MW_L^r\) of \(2\pi\)-periodic functions, where the variation of \(f^{(r-1)}\) in \([0,2\pi]\) is bounded by \(M\).