Trigonometric diameters of the Sobolev classes \(W^ r_ p\) in the space \(L_ q\) (Q1091553)

The estimates for the trigonometric n-diameters of the Sobolev class \(W^ r_ p\) (a subclass of \(L_ p\)-space, the rth derivative of whose elements also belongs to \(L_ p\) and has \(L_ p\)-norm \(\leq 1)\) in the space \(L_ q\) are given. Various results are obtained and, in particular, it is shown that with suitable restrictions on parameters p, q and r, the trigonometric diameter is order equivalent to the Kolmogorov diameter and the linear trigonometric diameter is order equivalent to the linear diameter.