Construction of $J^{\text{th}}$-stage Nonuniform Wavelets on Local Fields

Shah and Abdullah [Complex Analysis Operator Theory, 9 (2015), 1589-1608] have introduced a generalized notion of nonuniform multiresolution analysis (NUMRA) on local field $K$ of positive characteristic in which the translation set $\mathrm{\Lambda }$ acting on the scaling function to generate the core space $V_0$ is no longer a group, but is the union of $\mathcal{Z}$ and a translate of $\mathcal{Z}$, given by $\mathrm{\Lambda }=\{0,u(r)\mathrm{/}N\}+\mathcal{Z}$, where $N\ge 1$ is an integer and $r$ is an odd integer such that $r$ and $N$ are relatively prime, and $\mathcal{Z}=\{u(n):n\in {\mathbb{N}}_0\}$ is a complete list of distinct cosets of the unit disc $\mathfrak{D}$ in $K^{+}\mathrm{.}$ In this paper, we focus on the extension of nonuniform continuous wavelets to the construction of $J^{\text{th}}$-stage nonuniform discrete wavelets on local fields. We establish some general characterizations for the $J^{\text{th}}$-stage nonuniform discrete wavelet systems to be orthornormal bases in $L^2(\mathrm{\Lambda })$. Moreover, we establish a relation between the continuous wavelets of $L^2(K)$ and their discrete counterparts of $l^2(\mathrm{\Lambda })$.