Editing Haar wavelet (section)

== Haar functions and Haar system ==
For every pair ''n'', ''k'' of integers in <math>\mathbb{Z}</math>, the '''Haar function''' ''ψ''<sub>''n'',''k''</sub> is defined on the [[real line]] <math>\mathbb{R}</math> by the formula
:<math> \psi_{n,k}(t) = 2^{n / 2} \psi(2^n t-k), \quad t \in \mathbb{R}.</math>
This function is supported on the [[Semi-open interval|right-open interval]] {{nowrap| ''I''<sub>''n'',''k''</sub> {{=}}}} {{nowrap|[ ''k''2<sup>&minus;''n''</sup>, (''k''+1)2<sup>&minus;''n''</sup>)}}, ''i.e.'', it [[Zero of a function|vanishes]] outside that interval. It has integral 0 and norm&nbsp;1 in the [[Hilbert space]]&nbsp;[[Lp space|''L''<sup>2</sup>(<math>\mathbb{R}</math>)]],
:<math> \int_{\mathbb{R}} \psi_{n, k}(t) \, d t = 0, \quad \|\psi_{n, k}\|^2_{L^2(\mathbb{R})} = \int_{\mathbb{R}} \psi_{n, k}(t)^2 \, d t = 1.</math>
The Haar functions are pairwise [[Orthogonality#Orthogonal functions|orthogonal]],
:<math> \int_{\mathbb{R}} \psi_{n_1, k_1}(t) \psi_{n_2, k_2}(t) \, d t = \delta_{n_1n_2} \delta_{k_1k_2}, </math>
where <math>\delta_{ij}</math> represents the [[Kronecker delta]]. Here is the reason for orthogonality: when the two supporting intervals <math>I_{n_1, k_1}</math> and <math>I_{n_2, k_2}</math> are not equal, then they are either disjoint, or else the smaller of the two supports, say <math>I_{n_1, k_1}</math>, is contained in the lower or in the upper half of the other interval, on which the function <math>\psi_{n_2, k_2}</math> remains constant. It follows in this case that the product of these two Haar functions is a multiple of the first Haar function, hence the product has integral&nbsp;0.

The '''Haar system''' on the real line is the set of functions
:<math> \{ \psi_{n,k}(t) \; : \; n \in \mathbb{Z}, \; k \in \mathbb{Z} \}.</math>
It is [[Orthonormal basis|complete]] in ''L''<sup>2</sup>(<math>\mathbb{R}</math>): ''The Haar system on the line is an orthonormal basis in'' ''L''<sup>2</sup>(<math>\mathbb{R}</math>).