Haar system

Summary

The Haar system is an orthonormal basis of $L^2([0, 1])$. It consists of the constant function $1$ and the rescaled, shifted copies of one step function. This step function is $1$ on the left half and $-1$ on the right half of $[0, 1)$. Alfréd Haar introduced it in 1910, as an orthonormal system in which every continuous function has a uniformly convergent expansion. It is the simplest example of a wavelet basis, and it is a Schauder basis of $L^p([0, 1])$ for $1 \le p < \infty$.