Haar wavelet ($\psi ={\chi }_{[0,1/2)}-{\chi }_{[1/2,1)}$)

The Haar wavelet has one vanishing moment and compact support of unit length. Its scaling function is ${\chi }_{[0,1)}$, and its refinement coefficients are $h_0=h_1=1/\sqrt{2}$. The approximation spaces consist of functions constant on dyadic cells. Its discontinuity makes it effective for jump localization but limits smooth-function approximation.