Some generalized Besov-type space $b_{p\theta}^{\varphi}([0,1];h)$ with the Haar basis

We introduce the generalized Besov-type space $ B_{p\theta}^{\varphi}([0,1];H)$ over the Haar basis. We give the two-sided estimate for the norm of functions of the space in terms of their Fourier–Haar coefficients. Also, we establish a criterion for the embedding $B_{p\theta}^{\varphi}([0,1];H) \hookrightarrow L_{q\tau}[0,1]$ and some two-sided estimate for the approximation of $B^{\varphi}_{p\theta}([0,1],H)$ in the metric of $L_{q\tau}[0,1]$, with $1\leq p<q<+\infty$ and ${1\leq\tau<+\infty}$.