Construction of Jth-stage Nonuniform Wavelets on Local Fields

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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 Λ acting on the scaling function to generate the core space V₀ is no longer a group, but is the union of Z and a translate of Z, given by Λ={0,u(r)/N }+Z, where N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime, and Z={u(n): n₀} is a complete list of distinct cosets of the unit disc D in K+. In this paper, we focus on the extension of nonuniform continuous wavelets to the construction of Jth-stage nonuniform discrete wavelets on local fields. We establish some general characterizations for the Jth-stage nonuniform discrete wavelet systems to be orthornormal bases in L²(Λ). Moreover, we establish a relation between the continuous wavelets of L²(K) and their discrete counterparts of l²(Λ).