A localization property for ${\displaystyle B^s\_\left\{pq\right\}}$ and ${\displaystyle F^s\_\left\{pq\right\}}$ spaces

Let ${\displaystyle f^j=\sum _k{a}_{k}f(2^{j+1}x-2k)}$, where the sum is taken over the lattice of all points k in ${\displaystyle {ℝ}^n}$ having integer-valued components, j∈ℕ and ${\displaystyle a_k\in ℂ}$. Let ${\displaystyle A_{pq}^s}$ be either ${\displaystyle B_{pq}^s}$ or ${\displaystyle F_{pq}^s}$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${\displaystyle {ℝ}^n}$. The aim of the paper is to clarify under what conditions ${\displaystyle \parallel f^j\mid A_{pq}^s\parallel }$ is equivalent to ${\displaystyle 2^{j(s-n/p)}{\left(\sum _k{\left|a_k\right|}^p\right)}^{1/p}\parallel f\mid A_{pq}^s\parallel }$.