$$p$$-Adic Scaling and Generalized Scaling Sets

This article aims to develop MRA theory along with wavelet theory through corresponding sets in $$L^{2}(\mathbb{Q}_p)$$ . Generalized scaling sets are important in wavelet theory because they determine (multi)wavelet sets. Although, the theory of scaling sets and generalized scaling sets on $$\mathbb{R}$$ and local fields of positive characteristics are already developed to some extent, but it is yet to be studied on local fields of zero characteristic like $$\mathbb{Q}_p$$ . This article presents necessary conditions for scaling sets with counting formulae for the elements in scaling sets, and characterization of generalized scaling sets with examples.