Given the scaling equation of an multiresolution analysis, namely $$\varphi=\sqrt{2}\sum\limits_{k\in \mathbb{Z}}^{}h_k\varphi(2\cdot -k)$$ in $L²(ℝ)$-sense. Does this equation also hold on the fourierside, d.h. after used the Fouriertransform in $L²$ in L²-sense? Because using Fouriertransform one gets
\begin{align*} \hat{\varphi}&=\sqrt{2}\lim\limits_{n\to \infty} \sum\limits_{k=-n}^{n}h_k \hat{\varphi(2\cdot -k)}\\ &=\frac{\sqrt{2}}{2}\sum\limits_{k\in \mathbb{Z}}^{}h_k e^{-\frac{-ik\cdot }{2}}\hat{\varphi}\left(\frac{\cdot }{2}\right)\\ &=H\left(\frac{\cdot }{2}\right)\hat{\varphi}\left(\frac{\cdot }{2}\right). \end{align*}