Wavelet Expansions on the Cantor Group

The wavelet expansions in $L^p$-spaces on a locally compact Cantor group $G$ are studied. An order-sharp estimate of the wavelet approximation of an arbitrary function $f\in L^p(G)$ for $1\leqslant p<\infty$, in terms of the modulus of continuity of this function is obtained, and a Jackson–Bernstein type theorem on the approximation by wavelets of functions from the class $\operatorname{Lip}^{(p)}(\alpha;G)$ is proved.