 Solution

The answer is no with the integer-translation and dilation conventions of this multiresolution analysis. The orthonormal basis axiom forces $V_0$ to consist of $L^2$ functions constant on the cells $[k-1/2,k+1/2)$. Dilation forces $V_1$ to consist of functions constant on $[k/2-1/4,k/2+1/4)$. However, the proposed scaling function changes value at $1/2$, in the interior of the $V_1$ cell $[1/4,3/4)$. It is therefore not in $V_1$, contradicting nesting:Changing values at endpoints makes no difference in $L^2$. The usual Haar wavelet instead uses the scaling function ${\chi }_{[0,1)}$; an arbitrary half-unit translation does not preserve the required refinement grid.