Test students' ability to construct compactly supported orthonormal wavelet bases in L^2(R) with prescribed vanishing moments: derive finite-length scaling and wavelet filter coefficients from two-scale relations, prove orthonormality and compact support, verify the specified number of vanishing moments, analyze recurrence relations and their stability, compute and interpret frequency localization (Fourier decay and passband/stopband behavior), and relate these properties to multiresolution analysis and practical filter-bank implementation; include design exercises (e.g., Daubechies-type filters of given order), worked verification, and short proofs.