= Wavelet polynomial annihilation
{title2=$c_\psi(p;a,b)=0\quad(\deg p<q)$}
For $a>0$ and $b\in\mathbb R$, define the shifted, scaled analyzing <function> and its coefficient by
$$
\psi_{a,b}(x)=a^{-1/2}\psi\!\left(\frac{x-b}{a}\right),\qquad c_\psi(f;a,b)=\int_{\mathbb R}f(x)\overline{\psi_{a,b}(x)}\,dx=\sqrt a\int_{\mathbb R}f(b+au)\overline{\psi(u)}\,du.
$$
Assume the product is <integrable>; for example this holds for a <locally integrable function> $f$ and a bounded analyzing <function> with <compact support>. The bar is <complex conjugation>. This normalization preserves the <L2 norm> of the analyzing <function>. If $p$ is a <polynomial> of <degree of a polynomial> less than $q$, then $p(b+au)$ is a sum of <monomials> $u^r$ with $r<q$. <Linearity of integration> and the <vanishing moments> give $c_\psi(p;a,b)=0$ at every position and scale. Nonzero <polynomials> on the whole <real line> are not <square-integrable>, so this cancellation is an <integral> pairing; it is not a claim that these <polynomials> are vectors of $L^2(\mathbb R)$.
Here is a quantitative reason that <smooth> regions yield small details. Let $ψ$ have <compact support> and $q$ <vanishing moments>, and suppose $f$ is $q$ times <continuously differentiable> on an <interval> containing $b$ and $b+a\operatorname{supp}\psi$. Subtract its degree-$(q-1)$ <Taylor polynomial> at $b$. If $|f^{(q)}|\leq M$ on that <interval>, the <Taylor remainder> bound gives
$$
|c_\psi(f;a,b)|\leq\frac{M}{q!}a^{q+1/2}\int_{\mathbb R}|u|^q|\psi(u)|\,du.
$$
The factor $a^q$ comes from the <Taylor remainder>; $a^{1/2}$ comes from the coefficient normalization. Thus the analyzing <function> measures the departure from the local <polynomial> trend, rather than the height of $f$ itself. The <wavelet coefficient decay for Hölder functions> uses the same argument with a <Hölder-Taylor remainder bound>. These estimates concern regularity of $f$ on the analyzing <function>'s entire <support of a function>; extra <vanishing moments> alone do not make $ψ$ <smooth> or establish an <orthonormal wavelet> construction.<A Wavelet Tour of Signal Processing by Stéphane Mallat/Wavelet Zoom>[Wavelet Zoom, §6.1.1, printed p. 141, equations (6.1)–(6.2); §6.1.2, printed p. 143, Polynomial Suppression, equations (6.6)–(6.8)]{ref}
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