= Integer-frequency zeros of a scaling function
Assume the <scaling function> <Fourier transform> and the <MRA low-pass filter> are $C^{p-1}$, and $m^{(k)}(\pi)=0$ for $k<p$. For every nonzero <integer> $j$, write $j=2^r\ell$ with $\ell$ odd. Iterating the <scaling refinement equation> at $2\pi j+t$ supplies a factor $m(\pi\ell+t/2^{r+1})$. All its <derivatives> of order less than $p$ vanish at $t=0$, by periodicity. The <product rule> gives $\widehat\varphi^{(k)}(2\pi j)=0$ for $k<p$.
Back to article page
