= Low-pass filter of a multiresolution analysis
{title2=$m(\xi)$}
= MRA low-pass filter
{c}
{synonym}
For an <orthonormal> <scaling function> satisfying $\varphi(x)=\sqrt2\sum_kh_k\varphi(2x-k)$, its low-pass <Fourier series> symbol is $m(\xi)=2^{-1/2}\sum_kh_ke^{-ik\xi}$. The <scaling refinement equation> becomes $\widehat\varphi(2\xi)=m(\xi)\widehat\varphi(\xi)$. The symbol is initially an almost-everywhere defined periodic <function>; the assumption that the <scaling function> is <Lebesgue integrable> alone does not make it <smooth>. The <orthonormal> translates give the <quadrature mirror filter> identity.
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