Let the function $\eta\left(\frac{\cdot - m_ih}{h\sqrt{D}}\right)$, where $i=1,\ldots,N$, $N=\lceil \frac{1}{h}\rceil$, $h,D>0$ be rapidly decaying, i.e., it satisfies the inequality $|\eta(x)| \le A (1+|x|)^{-K}$ for all $x \in \mathbb{R}^n$, $A>0, K>n$. The notation $\lceil \cdot \rceil$ is the Ceiling function.
Let $A_{i,j} = \langle \eta_i(\cdot),\eta_j(\cdot)\rangle$, for $i,j=1,\ldots,N$, and let $A$ be the corresponding matrix associated with $A_{i,j}$, where $\eta_i(t)=\eta\left(\frac{t - m_ih}{h\sqrt{D}}\right)$ and $\eta_j(t)=\eta\left(\frac{t - m_jh}{h\sqrt{D}}\right)$. Let $y(\cdot)= \begin{pmatrix}\eta_1(\cdot),&\ldots&,\eta_N(\cdot)\end{pmatrix}^{T}$
I have the following expression for which I want a crude bound $$\sup_{t \in [0,M]}\sqrt{\langle A^{-1}y(t),y(t)\rangle} \le \sqrt{\sum_{i,j=1}^{\lceil \frac{1}{h}\rceil}(A^{-1})_{i,j}}\|y\|_{\infty}$$ I don't know what to do after this step, I think it can be crudely bounded by $\mathcal{O}(1/h)$, but I'm not sure. The entries of the matrix $A^{-1}$ depend on $h.$ Any help in this direction will be appreciated.