On various approaches to Besov-type spaces of variable smoothness

The paper is concerned with Besov spaces of variable smoothness $B^{varphi_{0}}_{p,q}(mathbb{R}^{n},{t_{k}})$, in which the norms are defined in terms of convolutions with smooth functions. A relation is found between the spaces $B^{varphi_{0}}_{p,q}(mathbb{R}^{n},{t_{k}})$ and the spaces $widetilde{B}^{l}_{p,q,r}(mathbb{R}^{n},{t_{k}})$, which were introduced earlier by the author.