Assess students' ability to construct a compactly supported orthonormal wavelet basis on R with a prescribed integer N of vanishing moments: derive a finite-length scaling (low-pass) filter h satisfying the two-scale relation and quadrature mirror/filter orthogonality conditions, compute the associated scaling and wavelet functions, prove orthonormality of integer translates and the multiresolution structure, establish regularity (smoothness) estimates of the scaling function via sum rules/strang-fix or spectral radius arguments, and verify directly that the resulting wavelet has N vanishing moments. Require both a general proof framework and an explicit worked example for a chosen N.