On Sobolev and Besov Spaces of Hybrid Regularity

Abstract The present article is concerned with the nonlinear approximation of functions in the Sobolev space $$H^{q}$$ H q with respect to a tensor-product, or hyperbolic wavelet basis on the unit $$n$$ n -cube. Here, $$q$$ q is a real number, which is not necessarily positive. We derive Jackson and Bernstein inequalities to obtain that the approximation classes contain Besov spaces of hybrid regularity. Especially, we show that all functions that can be approximated by classical wavelets are also approximable by tensor-product wavelets at least at the same rate. In particular, this implies that for nonnegative regularity, the classical Besov spaces of regularity $$B^{q+sn, \tau }_{\tau }$$ B τ q + s n , τ , with $$\nicefrac {1}{\tau } = s +\nicefrac {1}{2}$$ 1 τ = s + 1 2 , are included in the Besov spaces of hybrid regularity $$\mathfrak {B}^{q,s,\tau }_{\tau }$$ B τ q , s , τ , with isotropic regularity $$q$$ q and additional mixed regularity $$s$$ s .