= Haar wavelet
{c}
{title2=$\psi=\chi_{[0,1/2)}-\chi_{[1/2,1)}$}
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The Haar wavelet is a signed contrast between two adjacent half-intervals:
$$
\psi(x)=
\begin{cases}
1,&0\leq x<1/2,\\
-1,&1/2\leq x<1,\\
0,&\text{otherwise}.
\end{cases}
$$
It has <compact support>, zero <integral>, and unit <L2 norm>: $\int\psi=1/2-1/2=0$ and $\int|\psi|^2=1$. The zero integral is its one <vanishing moment>. Its <Haar scaling function> is $\phi=\mathbf1_{[0,1)}$, which records a coarse value rather than a contrast. Endpoint conventions do not alter the <L2 inner products>.
The normalized copies $\psi_{j,k}(x)=2^{j/2}\psi(2^jx-k)$ compare the halves of the <dyadic interval> $[k2^{-j},(k+1)2^{-j})$. Narrower copies are taller so that their <L2 norms> stay one. Thus a coefficient identifies both a location, through $k$, and a scale, through $j$.
\Image[/haar-wavelet-localized-contrasts.png]
{title=The Haar wavelet and two finer localized contrasts have zero integral and unit L2 norm}
For a simple <step function> $f$ equal to $2$ on $[0,1/2)$ and $6$ on $[1/2,1)$, its coarse coefficient is $\langle f,\phi\rangle=4$, while its first contrast is $\langle f,\psi\rangle=-2$. Thus \b[$f=4\phi-2\psi$]. The negative coefficient raises the second half and lowers the first. Finer contrasts vanish because $f$ is constant on each half. A jump aligned with this <dyadic interval> boundary therefore needs only two nonzero coefficients; jumps at other positions can require details at many scales.
Its <jump discontinuities> make the <Haar wavelet> particularly transparent for changes between neighbouring regions, while limiting approximation of smooth variation. A <Daubechies wavelet> uses a more elaborate pattern to obtain additional <vanishing moments> and, at sufficiently high order, smoother basis functions. The elementary <Haar refinement identity> has coefficients $h_0=h_1=1/\sqrt2$ and explains how the coarse and fine descriptions fit together.
The historical source is Alfred Haar's https://history-of-approximation-theory.com/fpapers/haar1910.pdf[Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen 69 (1910), pp. 331–371]{ref}. Chapter III, beginning on page 361, constructs the constant function and these localized piecewise-constant orthogonal functions. Its original endpoint assignments differ from the half-open convention used here, without changing <L2 inner products>.
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