Characterization of low-pass filters on local fields of positive characteristic (Q2827406)

Let \(K\) be a local field of positive characteristic, such as the Cantor dyadic group and the Vilenkin \(p\)-group. Multiresolution analysis \(\{V_j: \, j \in \mathbb Z\}\) of \(L^2(K)\) was introduced by \textit{H. Jiang} et al. [J. Math. Anal. Appl. 294, No. 2, 523--532 (2004; Zbl 1045.43012)], where \(V_j\) are nested closed subspaces of \(L_2(K)\). A function \(\varphi \in L^2(K)\) is called a pre-scaling function, if its translates form a Riesz basis of \(V_0\). In this paper, the author presents necessary and sufficient conditions that a function \(m\) is a low-pass filter associated with a pre-scaling function \(\varphi\).