= Orthonormal wavelet
{title2=$\psi_{j,k}(x)=2^{j/2}\psi(2^jx-k)$}
{wiki}
An orthonormal wavelet $\psi$ generates an <orthonormal basis> of $L^2(\mathbb R)$ through
$$
\psi_{j,k}(x)=2^{j/2}\psi(2^jx-k),\qquad j,k\in\mathbb Z.
$$
The factor $2^{j/2}$ compensates for the shrinking <support of a function>, keeping the <L2 norm> equal to one. <Orthogonality> makes different coefficients separate squared-norm contributions, and completeness allows the <wavelet series> to recover any <square-integrable function> in the <L2 norm>.
For the <Haar wavelet>, wavelets at one scale have disjoint <supports>. At different scales their <dyadic intervals> are either disjoint or nested. On the support of the finer wavelet, the coarser one is constant, while the finer one has zero <integral>, so their <L2 inner product> is zero. Together with the unit <L2 norm>, this explains <orthonormality>. On the bounded <interval> $[0,1]$, add the constant <Haar scaling function> and use $j\geq0$, $0\leq k<2^j$; the constant function is not an $L^2(\mathbb R)$ basis element on the whole line.
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