= Wavelet completion of a multiresolution filter
For a <scaling refinement equation> $\phi=\sum_na_n\phi(2\cdot-n)$, set $m(t)=\frac12\sum_na_ne^{-int}$. The <quadrature mirror filter> identity makes the two rows $(m(t),m(t+\pi))$ and $(q(t),q(t+\pi))$ a unitary matrix, where $q(t)=-e^{-it}\overline{m(t+\pi)}$. The complementary coefficients $b_n=(-1)^n\overline{a_{1-n}}$ yield $\psi=\sum_nb_n\phi(2\cdot-n)$. In a <multiresolution analysis>, coefficient-space completeness shows its translates form an <orthonormal basis> of $V_1\ominus V_0$, hence its dilations and translates form an <orthonormal wavelet> basis of the whole space.
Back to article page
