= Scaling refinement equation
{title2=$\varphi(x)=\sqrt2\sum_kh_k\varphi(2x-k)$}
Nesting in an orthonormal <multiresolution analysis> expresses a <scaling function> in the finer-scale <orthonormal basis>. The associated low-pass symbol is $m(\xi)=2^{-1/2}\sum_kh_ke^{-ik\xi}$, and $\widehat\varphi(2\xi)=m(\xi)\widehat\varphi(\xi)$. If $\varphi$ has <compact support>, only finitely many coefficients $h_k$ are nonzero. With $\widehat\varphi(0)=1$, iteration gives $\widehat\varphi(\xi)=\prod_{j\ge1}m(\xi/2^j)$ when the limit exists.
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