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 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}.
