Let $x:=(x_{i,j})_{i\in \mathbb{N},\, j=0,\dots,2^i}$ be a real-sequence and consider the (small) [Besov-type sequence spaces][1] with quasi-norms for $0<q,p,\alpha< \infty$
$$
\|x\|_{\alpha,p,q}
:=
\left(
\sum_{i=0}^{\infty}
2^{\alpha i p}
\biggl(
\sum_{j=0}^{2^i}|x_{i,j}|^p2^{-i\,p}
\biggr)^{q/p}
\right)^{1/q}
.
$$
If $0<\alpha<\tilde{\alpha}<\infty$ and $\|x\|_{\tilde{\alpha},p,q}<\infty$ then were can I find estimates on the tail-sum
$$
\sum_{i=I}^{\infty} 
2^{\alpha i p}
\biggl(
\sum_{j=0}^{2^i}|x_{i,j}|^p2^{-i\,p}
\biggr)^{q/p}
$$
representing the norm $\|\cdot\|_{\alpha,p,q}$ of the approximation of $x$ by $(x_{i,j})_{i=0,j=0,\dots,2^i}^I$.
What I'm basically asking is can we do better than
$$
\label{1}
\tag{1}
\sum_{i=I}^{\infty} 
2^{\alpha i p}
\biggl(
\sum_{j=0}^{2^i}|x_{i,j}|^p2^{-i\,p}
\biggr)^{q/p}
\lesssim 
\sum_{i=I}^{\infty} 
2^{(\alpha-1) i p}
2^{(1-\tilde{\alpha})ip}
=
\frac{2^{(\alpha - \tilde{\alpha})p I}}{1 - 2^{(\alpha - \tilde{\alpha})p}}
\in \mathcal{O}_{p,\alpha,\tilde{\alpha}}\Big(
2^{(-\tilde{\alpha}+\alpha)pI}
\Big)?
$$

---

I ask since this estimate is *crude* in the sense that, I only note that if $\|x\|_{\tilde{\alpha},p,q}<\infty$ then we must have
$$
2^{\alpha i p} 
\biggl(
\sum_{j=0}^{2^i}|x_{i,j}|^p2^{-i\,p}
\biggr)^{q/p}
<C
$$
for some constant $C>0$ not depending on $i$; whence we get the bound
$$
\biggl(
\sum_{j=0}^{2^i}|x_{i,j}|^p2^{-i\,p}
\biggr)^{q/p}
\lesssim
2^{-\alpha i p}
$$
which is the only thing I used in \eqref{1}.

 [1]: https://link.springer.com/article/10.1007/s00365-010-9115-6