Implementing Daubechies wavelet transform with weighted finite automata

We show that the compactly supported wavelet functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $W_2, W_4, W_6, \ldots$\end{document} discovered by Daubechies [6] can be computed by weighted finite automata (WFA) introduced by Culik and Karhumäki [2]. Furthermore, for 1-D case, a fixed WFA with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $2^n + n(N-2)$\end{document} states can implement any linear combination of dilations and translations of a basic wavelet \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $W_N$\end{document} at resolution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $2^n$\end{document}. The coefficients of the wavelet transform specify the initial weights in the corresponding states of the WFA. An algorithm to simplify this WFA is presented and can be employed to compress data. It works especially well for smooth and fractal-like data.