Solve the following problem. Show step-by-step with explanation.

Consider the Haar wavelet and the discretization 70 = 1 and no = 2. As we see in chapter 4, the associated discrete wavelet functions are an orthonormal basis in L2(R). Thus, they form a tight frame with A = B = 1. Show that dw = In 2 W where V (w) is the Fourier transform of the Haar mother wavelet. Check that this is consistent with the result of theorem 53. Theorem 53. If the discretized wavelet functions 4mn(t) of equation (3.85), for both positive and neg- ative values of the scale parameter no, form a frame then the corresponding mother wavelet up(t) satis- fies the admissibility condition (3.55) through the inequalities 4/mn (t) = 1. nm (t) = no" -m/2 4 (not - nTo) , m,ne Z, (3.85) DO |wi-14 ( w) 12 dw