On various approaches to Besov-type spaces of variable smoothness

The paper is concerned with Besov spaces of variable smoothness $B_{p,q}^{{\phi }_0}({\mathbb{R}}^n,\{t_k\})$, in which the norms are defined in terms of convolutions with smooth functions. A relation is found between the spaces $B_{p,q}^{{\phi }_0}({\mathbb{R}}^n,\{t_k\})$ and the spaces ${\tilde{B}}_{p,q,r}^l({\mathbb{R}}^n,\{t_k\})$, which were introduced earlier by the author.