p-Adic Scaling Set and Generalized Scaling Set
Debasis Haldar
TL;DR
The paper advances p-adic multiresolution analysis by developing the concepts of scaling sets and generalized scaling sets within L^2(Q_p). It defines scaling sets, establishes their measure properties under dilation, and derives the refinable framework with a scaling filter m0 that governs the Fourier relation of scaling functions. It then extends to generalized scaling sets of order L, showing that they can be characterized via unions of dilates of multiwavelet sets and providing concrete examples linked to Kozyrev and related constructions. These results deepen the understanding of p-adic wavelet theory and offer a structured pathway to construct MRA and multiwavelets on Q_p with explicit sets and filters, enhancing both theory and potential applications in p-adic harmonic analysis.
Abstract
The main goal of this paper is to develop the MRA theory along with wavelet theory in L2(Qp). Generalized scaling sets are important in wavelet theory because it determine multiwavelet sets. Although the theory of scaling set and generalized scaling set on R and many other local field of positive characteristic are available but not on Qp. This article contains discussion of some necessary conditions of scaling set and characterize generalized scaling set with examples.