Nonuniform Multiresolution Analysis on Local Fields of Positive Characteristic

Abstract

A generalization of Mallat’s classic theory of multiresolution analysis (MRA) on local fields of positive characteristic was considered by Jiang et al. (J Math Anal Appl 294:523–532, 2004). In this paper, we present a notion of nonuniform MRA on local field $K$ of positive characteristic. The associated subspace $V_0$ of $L^2(K)$ has an orthonormal basis, a collection of translates of the scaling function $\varphi$ of the form $\{ \varphi (x-\lambda ) \}_{ \lambda \in \Lambda }$ where $\Lambda = \{ 0,r/N \}+ \mathcal{Z}, \,N \ge 1$ is an integer and $r$ is an odd integer such that $r$ and $N$ are relatively prime and $\mathcal{Z}=\{u(n): n\in \mathbb {N}_{0}\}$. We obtain the necessary and sufficient condition for the existence of associated wavelets and present an algorithm for the construction of nonuniform MRA on local fields starting from a low-pass filter $m_{0}$ with appropriate conditions.