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Use localization and cancellation of the assumed interval-adapted wavelet basis: at scale $j$, every detail wavelet has support diameter at most $B{2}^{-j}$, L1 norm at most $B^{\prime }{2}^{-j/2}$, and annihilates polynomials of degree less than $q$. These bounds also apply to the modified boundary wavelets; a restriction of an arbitrary whole-line wavelet without the cancellation-preserving boundary construction would not suffice. The finitely many coarse scaling functions can be treated separately.

Write $\alpha =r+\beta$ with $0<\beta \le 1$ as above. Because $\alpha <q$, we have $r<q$. Choose $x_0$ in the support of a function of ${\psi }_{j,n}$, and let $T_{x_0}$ be the degree-$r$ Taylor polynomial of $f$. The Hölder-Taylor remainder bound givesFor $r=0$ this is the Hölder seminorm bound. For $r\ge 1$, the integral Taylor remainder is bounded using $f^{(r)}(t)-f^{(r)}(x_0)$, whose magnitude is at most $[f^{(r)}{]}_{C^{0,\beta }}|t-x_0{|}^{\beta }$.

The vanishing moments remove $T_{x_0}$ from the inner product. Applying the remainder estimate and the L1 norm bound on the localized support yields the wavelet coefficient decay for Hölder functionsThusHere $C$ depends on the fixed wavelet family, the exponent and boundary construction, but not on $f,j,n$. The proof uses the assumed regularity and moments; it does not identify the minimal Daubechies wavelet order with its differentiability order, which need not coincide.

Let $M$ bound the piecewise Hölder norms, and let $B=\Vert f{\Vert }_{\infty }$. At scale $j$, classify a wavelet as bad when its localized support of a function meets a discontinuity, and as good otherwise. Compact support and bounded overlap imply at most $C_{0}K$ bad wavelets per scale and at most $C_1{2}^j$ total wavelets. A good support lies in a single smooth piece, so the preceding wavelet coefficient decay for Hölder functions gives $|c_{j,n}|\le CM{2}^{-j(\alpha +1/2)}$. The permitted bounded-function estimate gives $|c_{j,n}|\le CB{2}^{-j/2}$ for bad coefficients.

Consider a specific N-term approximation: retain all coarse scaling functions and all wavelets through scale $J$, and also retain the bad coefficients at scales $J<j\le L$, where $L=\lceil 2\alpha J\rceil$. Since $\alpha >1/2$, $L\ge J$ for large $J$. Its term count is bounded byBy the Parseval identity, its squared L2 norm error is the sum of omitted squared coefficients. The good tail satisfiesand the remaining bad tail satisfiesChoose $J=\lfloor {\log }_2(N/(2C_2^{\prime }))\rfloor$ with a fixed sufficiently large $C_2^{\prime }$; then the $O(KJ)$ extra terms fit the remaining budget for all sufficiently large $N$, while $2^J$ is comparable to $N$. Padding to $N$ terms does not increase the error. The best N-term approximation is at least as accurate as this constructed selection, soThe constant may depend on the finite number of jumps, the piecewise Hölder norms, $B$ and the fixed wavelet family. Values exactly at the jumps are immaterial in $L^2$. The improvement comes from retaining only a bounded number of jump-crossing coefficients at each finer scale, as in best N-term wavelet approximation of piecewise Hölder functions.

For $a>0$ and $b\in \mathbb{R}$, define the shifted, scaled analyzing function and its coefficient byAssume 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 $\psi$ have compact support and $q$ vanishing moments, and suppose $f$ is $q$ times continuously differentiable on an interval containing $b$ and $b+a\mathrm{supp}\psi$. Subtract its degree-$(q-1)$ Taylor polynomial at $b$. If $|f^{(q)}|\le M$ on that interval, the Taylor remainder bound givesThe 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 $\psi$ smooth or establish an orthonormal wavelet construction.^[ref]