On various approaches to Besov-type spaces of variable smoothness

Abstract: 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.