Nonlinear approximation of function spaces of mixed smoothness

We study the approximative properties of $L_q$-greedy algorithms with respect to the wellknown system $U^d$ of shifts of Dirichlet kernels on the Nikol'skiĭ–Besov classes $\mathrm{SB}^r_{p\theta}(\mathbb T^d)$ and the Lizorkin–Triebel classes $\mathrm{SF}^r_{p\theta}(\mathbb T^d)$ of functions of mixed smoothness.