  We give a complete description of spectrum of the wavelet Galerkin operator
$$R_{m_0,m_0}f(z)=\frac{1}{N}\sum_{w^N=z}|m_0|^2(w)f(w),\quad(z\in\m
athbb{T})$$ associated to a a low-pass filter $m_0$ and a scale $N$, in the
Banach spaces $C(\mathbb{T})$ and $L^{p}(\mathbb{T})$, $1\leq p\leq\infty$.
