The Spectrum of the Wavelet Galerkin Operator

Abstract.

We give a complete description of spectrum of the wavelet Galerkin operator

$$ R_{m_0 ,{\kern 1pt} m_0 } f(z) = \frac{1} {N}\sum\limits_{w^N = z} {|m_0 |^2 (w)f(w),\quad (z \in \mathbb{T})} $$

associated to a low-pass filter m₀ and a scale N, in the Banach spaces $C(\mathbb{T})$ and $L^P (\mathbb{T}),{\kern 1pt} \,1 \leq p \leq \infty .$