Relative width of smooth classes of multivariate periodic functions with restrictions on iterated Laplace derivatives in the \(L_2\)-metric (Q2426786)

Let \(\Delta\) denote the usual Laplace operator in \(d\) dimensions and consider the class \(W_2(\Delta^r)\) of \(2\pi\)-periodic \(d\)-variate functions subject to the condition that \(\int_{[-\pi,\pi]^d} | \Delta^r f(x)| ^2 \,dx \leq 1\). The authors determine the asymptotic behaviour of the relative Kolmogorov \(n\)-width \[ K_n(W_2(\Delta^r), W_2(\Delta^r), L_q([-\pi,\pi]^d)). \]