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A Details on endpoint-interpolating B-spline wavelets A 2 Endpoint-interpolating linear B-spline wavelets This appendix presents the matrices required to apply endpoi Figure 9 shows a few typical scaling functions and wavelets for terpolating B-spline wavelets of low degree. (The Matlab code linear B-splines. The synthesis matrices Py and o for endpoint used to generate these matrices is available from the authors upon interpolating linear B-spline wavelets are uest.)These concrete examples should serve to elucidate the ideas presented in Section 3. To emphasize the sparse structure of the matrices, zeros have been omitted. Diagonal dots indicate that he previous column is to be repeated the appropriate number of times, shifted down by two rows for each column. The P matrices ave entries relating the unnormalized scaling functions defined in Section 3, while the g matrices have entries defining normalized minimally supported wavelets. Columns of the Q matrices that are not represented exactly with integers are given to six decimal places. A1 Haar wavelets The B-spline wavelet basis of degree 0 is simply the haar basis de- cribed in Section 2 of Part 1 amples of the Haar basis scal ing functions and wavelets are ed in Figure 8. The synthesis matrices P and O are 1009859 八 Figure 8 Piecewise-constant B-spline scaling functions and Figure9 Linear B-spline scaling functions and wavelets for =3A Details on endpoint-interpolating B-spline wavelets This appendix presents the matrices required to apply endpointinterpolating B-spline wavelets of low degree. (The Matlab code used to generate these matrices is available from the authors upon request.) These concrete examples should serve to elucidate the ideas presented in Section 3. To emphasize the sparse structure of the matrices, zeros have been omitted. Diagonal dots indicate that the previous column is to be repeated the appropriate number of times, shifted down by two rows for each column. TheP matrices have entries relating the unnormalized scaling functions defined in Section 3, while the Q matrices have entries defining normalized, minimally supported wavelets. Columns of the Q matrices that are not represented exactly with integers are given to six decimal places. A.1 Haar wavelets The B-spline wavelet basis of degree 0 is simply the Haar basis described in Section 2 of Part 1. Some examples of the Haar basis scaling functions and wavelets are depicted in Figure 8. The synthesis matrices Pj and Qj are Pj = 2 6 4 1 1 1 1 1 1 3 7 5 Qj = p 2j 2 2 6 4 1 1 1 1 1 1 3 7 5 Figure 8 Piecewise-constant B-spline scaling functions and wavelets for j = 3. A.2 Endpoint-interpolating linear B-spline wavelets Figure 9 shows a few typical scaling functions and wavelets for linear B-splines. The synthesis matrices Pj and Qj for endpointinterpolating linear B-spline wavelets are P1 = 1 2 h 2 1 1 2 i P2 = 1 2 " 2 1 1 2 1 1 2 # Pj3 = 1 2 2 6 6 6 6 6 4 2 1 1 2 1 1 2 1 1 2 1 1 2 3 7 7 7 7 7 5 Q1 = p 3 h 1 1 1 i Q2 = q 3 64 " 12 11 1 6 6 1 11 12 # Qj3 = q 2j 72 2 6 6 6 6 6 6 6 4 11. 022704 10. 104145 1 5. 511352 6 0. 918559 10 1 6 6 1 10 6 1 1 6 10 0. 918559 6 5. 511352 1 10. 104145 11. 022704 3 7 7 7 7 7 7 7 5 Figure 9 Linear B-spline scaling functions and wavelets forj = 3. 7