Abstract: In order to facilitate the use of wavelets, the study on frame wavelets (also called framelets) and framelet packets in digital setting has been addressed in recent years. An approach to construct a class of $J$-stage framelet packets for $\ell^2(\mathbb{Z})$ is given by Lu and Yi. However, the detailed results on how to use them are missing. To further improve the theoretical system of $J$-stage framelet packets for $\ell^2(Z)$, by following the work of Lu and Yi, the fast decomposition and reconstruction algorithms are given in this paper, with which one can establish the relationship of coefficients between different stages. For the convenience of using, the detailed data of framelet packets for $\ell^2(Z)$ when the number $n$ of mother framelet ranges from $1$ to $4$ are constructed. Finally, a numerical experiment is given to illustrate the perfect reconstruction of framelet packets.

Key words: framelet packets, convolution, first-stage framelets, sequence spaces