= MRA projection Fourier identity
{c}
{title2=$\|P_jf\|_2^2$}
Use $\widehat f(\xi)=\int f(x)e^{-ix\xi}\,dx$ and let $P_j$ be the <orthogonal projection> onto the closed span of the orthonormal translates and dilates of a <scaling function>. If $\widehat f$ is supported in $[-R,R]$ and $R<\pi2^j$, then
$$
\|P_jf\|_2^2=\frac1{2\pi}\int|\widehat f(\xi)|^2|\widehat\varphi(2^{-j}\xi)|^2\,d\xi.
$$
Indeed, the <Plancherel theorem> writes the coefficient against $\varphi_{j,k}$ as $2^{-j/2}(2\pi)^{-1}\int\widehat f(\xi)\overline{\widehat\varphi(2^{-j}\xi)}e^{ik2^{-j}\xi}\,d\xi$. With $\xi=2^j\theta$, these are $2^{j/2}$ times the <Fourier series> coefficients of $\widehat f(2^j\theta)\overline{\widehat\varphi(\theta)}$ on $[-\pi,\pi]$. The <Parseval identity> proves the formula. It shows that continuity and unit modulus at zero imply density of the refinement spaces, and conversely that density forces this unit modulus when the <Fourier transform> is continuous at zero.
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