I've been working with wavelets for a project, more specifically implementing the Fast Wavelet Transform, which requires the use of the dual high pass and low pass filters (for the forward FWT). In the literature (such as in Karsten Urban's "Wavelet Methods for Elliptic Partial Differential Equations,") they give the following relation for the class of Biorthogonal B-Spline wavelets:
$$ \tilde{b}_{k} = (-1)^k a_{1-k} $$
Where the $\tilde{b}_{k}$ are the refinement coefficients of the dual wavelet, and the $a_{k}$ are the refinement coefficients of the primal scaling function. These sequences of coefficients are known to be:
$$ \textbf{a} = \{\{-1, 0.5\}, \{0, 1\}, \{1, 0.5\}\}\\ \tilde{\textbf{b}} = \{\{0, -0.5\}, \{1, 1\}, \{2, -0.5\}\} $$
The first element of each pair is the index k, while the second is the actual value. However, there seems to be an inconsistency since, for instance, $\tilde{b}_{0} = -0.5$ and if we use the relation, we get $\tilde{b}_{0} = (-1)^0a_{1-0} = a_{1} = 0.5$ which doesn't match.
Does anyone know what I am missing, or if I am making an error? Thank you.