= Vanishing moment
{title2=$\int x^r\psi(x)\,dx=0$}
A <vanishing moment> is a cancellation condition: positive and negative parts of an analyzing <function> balance when weighted by a <monomial>. Having $q$ <vanishing moments> means that the first $q$ such balances hold, starting with the unweighted <integral>:
$$
\int_{\mathbb R}x^r\psi(x)\,dx=0,\qquad r=0,\ldots,q-1,\qquad q\geq1.
$$
Consequently a <wavelet> ignores local <polynomial> trends of <degree of a polynomial> less than $q$. A <smooth> feature is close to its <Taylor polynomial>, so only the small <Taylor remainder> contributes to its detail coefficient. A <jump discontinuity> cannot be approximated this way across its edge; its <jump response of a localized analyzing function> can remain large relative to details from a <smooth> region.<A Wavelet Tour of Signal Processing by Stéphane Mallat/Wavelet Bases>[Wavelet Bases, §7.2.1, printed p. 206, equation (7.69) and the paragraph explaining cancellation of Taylor polynomials]{ref}
These are ordinary absolutely convergent <integrals>, not a prescription for assigning values to divergent expressions. A sufficient assumption is
$$
\int_{\mathbb R}(1+|x|^{q-1})|\psi(x)|\,dx<\infty.
$$
It makes $ψ$ and all $x^r\psi$ for $r<q$ <integrable functions>. Being <square-integrable> alone does not ensure these weighted <integrals> exist. For a <square-integrable function> with <compact support>, the <Cauchy-Schwarz inequality> guarantees every such weighted <integral> is finite. Saying \b[exactly $q$] <vanishing moments> additionally requires an existing, nonzero next <integral> $\int x^q\psi(x)\,dx$; the usual phrase “has $q$” allows more cancellation.
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