We extend and improve the results of W. Lang (1998) on the wavelet analysis on the Cantor dyadic group C. Our construction is realized on a locally compact abelian group G which is defined for an integer p 2 and coincides with C when p = 2. For any integers p, n 2 we determine a function ϕ in L 2 (G) which 1) is the sum of a lacunary series by generalized Walsh functions, 2) has orthonormal "integer" shifts in L 2 (G), 3) satisfies "the scaling equation" with p n numerical coefficients, 4) has compact support whose Haar measure is proportional to p n , 5) generates a multiresolution analysis in L 2 (G). Orthogonal wavelets ψ with compact supports on G are defined by such functions ϕ. The family of these functions ϕ is in many respects analogous to the well-known family of Daubechies' scaling functions. We give a method for estimating the moduli of continuity of the functions ϕ, which leads to sharp estimates for small p and n. We also show that the notion of adapted multiresolution analysis recently suggested by Sendov is applicable in this situation.

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For any integers p, n ≥ 2 necessary and sufficient conditions are given for scaling filters with p n many terms to generate a p-multiresolution analysis in L 2 (R + ). A method for constructing orthogonal compactly supported p-wavelets on R + is described. Also, an adaptive p-wavelet approximation in L 2 (R + ) is considered.

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