Math 6420 • PDE

The Haar function \(\phi (x)\) is defined by \[ \phi (x)= \begin {cases} 1, & 0<x<1,\\ 0, & \text {elsewhere} \end {cases} \] and the mother Haar wavelet \(W(x)\) is defined by \[ W(x)= \begin {cases} 1, & 0<x<1/2,\\ -1, & 1/2<x<1,\\ 0, & \text {elsewhere}. \end {cases} \] Express the function \[ f(x)= \begin {cases} 2, & 0<x<1/4,\\ 0, & 1/4<x<1/2,\\ -1, & 1/2<x<3/4,\\ 1, & 3/4<x<1,\\ 0, & \text {elsewhere} \end {cases} \] as a linear combination of 4 orthogonal functions: the Haar function and 3 Haar wavelets \[ W_{mn}(x)=2^{m/2}W(2^mx-n), \] where \(m\) and \(n\) are appropriate integers.