I was wondering if there is some connection between positional notation for number systems and the Fourier transform. Here is what led to this proposition. Say we look at the first few integers in binary: $$00000_2 = 0_{10} \hspace{3cm} 00101_2 = 5_{10} $$ $$00001_2 = 1_{10} \hspace{3cm} 00110_2 = 6_{10} $$ $$00010_2 = 2_{10} \hspace{3cm} 00111_2 = 7_{10} $$ $$00011_2 = 3_{10} \hspace{3cm} 01000_2 = 8_{10} $$ $$00100_2 = 4_{10} \hspace{3cm} 01001_2 = 9_{10} $$
Then consider some functions $f_p : D \rightarrow \{1,0\}$ with $D=\{x_i \in \mathbb{Z} : x\ge 0 \}$. Further let the $f_p$'s be periodic, with a "frequency" $p$. These functions can be described in terms of a matrix $F_{px_i}=f_p(x_i)$
\begin{equation} F_{px_i}=\begin{bmatrix} 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & ... \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 0 & ...\\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & ...\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & ...\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & ...\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots &\ddots \\ \end{bmatrix} \end{equation} Then, for e.g., to get $2_{10}$ in binary we have $[0,1,0,...]^T=F\vec{v}_{2_{10}}$, (so first entry is the right most position) where $\vec{v}_{2_{10}}=[0, 0, 1, 0,...]^T$. This is kind of like a Fourier transform. Has this been formally explored somewhere?