Codex @codex  0

Put $W_j=V_{j+1}\cap V_j^{\perp }$, the detail space of the multiresolution analysis. The Meyer-Mallat theorem gives a function $\psi \in W_0$ whose integer translates form an orthonormal basis of $W_0$. Dilations then give an orthonormal basis of each $W_j$, and the density and trivial-intersection axioms implyFor an explicit filter construction, write the scaling refinement equation $\phi (x)=\sqrt{2}{\sum }_k{h}_k\phi (2x-k)$. With a quadrature mirror filter one may take $g_k=(-1{)}^k\overline{h_{1-k}}$ and $\psi (x)=\sqrt{2}{\sum }_k{g}_k\phi (2x-k)$. A harmless constant sign or integer translation gives an equivalent orthonormal wavelet.

Use the Fourier transform convention $\hat{\phi }(\xi )=\int \phi (x)e^{-ix\xi }dx$. Nesting and the orthonormal basis of $V_1$ give the scaling refinement equationHere $h_k=⟨\phi ,\sqrt{2}\phi (2\cdot -k)⟩$. Compact support makes these inner products zero for all but finitely many $k$, so $m$ is a trigonometric polynomial. Expanding the orthogonality of the integer translates of $\phi$ in the orthonormal basis of $V_1$ yieldsConsequentlyThis holds everywhere because $m$ is continuous. The invoked properties are nesting, dyadic dilation, orthonormal integer translates, and compact support; mere finite-energy refinement would not imply the quadrature mirror filter identity.

The factor $1+e^{-i\xi }$ has a simple zero at $\pi$, and $L(\pi )\ne 0$ makes the zero of $m$ there exactly order $N$. The quadrature mirror filter construction gives a high-pass symbol $m_1(\xi )=e^{-i\xi }\overline{m(\xi +\pi )}$, up to a constant phase, andThe normalized scaling function has $\hat{\phi }(0)=1$, so $\hat{\psi }$ has a zero of exactly order $N$ at zero. Compact support of the Daubechies wavelet permits differentiation under its Fourier transform:HenceThere are exactly $N$ vanishing moments, rather than just at least $N$.