Compute the first four terms of the wavelet series expansion of the function used in Examp...

Compute the first four terms of the wavelet series expansion of the function used in Example 7.7 with starting scale j0 = 1. Write the resulting expansion in terms of the scaling and wavelet functions involved. How does your result compare to the example, where the starting scale was j0 = 0?

EXAMPLE 7.7: The Haar wavelet series expansion of y = x2.

Consider the simple function

shown in Fig. 7.15(a). Using Haar wavelets—see Eqs. (7.2-14) and (7.2-30)— and a starting scale j0 = 0, Eqs. (7.3-2) and (7.3-3) can be used to compute the following expansion coefficients:

(7.2-14)

(7.2-30)

(7.3-2)

(7.3-3)

FIGURE 7.15 A wavelet series expansion of y = xusing Haar wavelets.

Substituting these values into Eq. (7.3-1), we get the wavelet series expansion

(7.3-1)

The first term in this expansion uses c0 (0) to generate a subspace V0approximation of the function being expanded. This approximation is shown in Fig. 7.15(b) and is the average value of the original function. The second term uses d0(0) to refine the approximation by adding a level of detail from subspace W0. The added detail and resulting V1approximation are shown in Figs. 7.15(c) and (d), respectively. Another level of detail is added by the subspace W1coefficients d1(0) and d1(1). This additional detail is shown in Fig. 7.15(e), and the resulting V2 approximation is depicted in 7.15(f). Note that the expansion is now beginning to resemble the original function. As higher scales (greater levels of detail) are added, the approximation becomes a more precise representation of the function, realizing it in the limit as j → ∞.