Wavelet Analysis: A notation problem in this paper

Question

Because of my research work, I am reading this paper: PDE net. But I ran into a notation problem: In definition 2.1, I do not understand how $\textbf{q}[ \textbf{k} ]$ yields a real number. Therefore, I could not verify the authors' claims about the order of sum rules of the Haar wavelet frame filter bank and also the filter \begin{equation*} \textbf{q} = \begin{pmatrix} 1 & 0 & -1 \\ 2 & 0 & -2 \\ 1 & 0 & -1 \end{pmatrix} \end{equation*} I checked their citations and some other literatures, but most of them just said $\{\textbf{q}[\textbf{k}]\}$ is a sequence of real numbers, which does not help me to understand the order of sum rules at all. I am grateful if anyone could give me some hint about this notation.