Orthogonal wavelets with compact support on locally compact Abelian groups

We extend and improve the results of W. Lang (1998) on the wavelet analysis on the Cantor dyadic group $\mathcal C$. Our construction is realized on a locally compact abelian group $G$ which is defined for an integer $p\geqslant2$ and coincides with $\mathcal C$ when $p=2$. For any integers $p,n\geqslant 2$ we determine a function $\varphi$ in $L^2(G)$ which

• is the sum of a lacunary series by generalized Walsh functions,
• has orthonormal “integer” shifts in $L^2(G)$,
• satisfies “the scaling equation” with $p^n$ numerical coefficients,
• has compact support whose Haar measure is proportional to $p^n$,
• generates a multiresolution analysis in $L^2(G)$.

Orthogonal wavelets $\psi$ with compact supports on $G$ are defined by such functions $\varphi$. The family of these functions $\varphi$ 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 $\varphi$, 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.