The scaling equation ( 7.194) implies that any autocorrelation filter verifies h =0 for n в‰ 0. Polynomials of degree 2 p в€’ 1 = 3 are interpolated by this function. The graph of this interpolation function is in Figure 7.19(b). For p = 2, the Deslauriers-Dubuc interpolation function П† is the autocorrelation of the Daubechies 2 scaling function, shown in Figure 7.10. ДЇor p = 1, П† o = 1 and П† are the piecewise linear tent functions with a support that. The autocorrelation П† is the Deslauriers-Dubuc interpolation function, which support.

Daubechies filters h o have 2 p nonzero coefficients and the resulting scaling function П† o has a support of size 2 p в€’ 1. The Daubechies theorem (7.7) designs minimum-size conjugate mirror filters h o that satisfy this condition. Since h = h o в‹† h ВЇ o, it follows that h ^ ( П‰ ) = | h ^ o ( П‰ ) | 2, and thus h ^ o ( П‰ ) has a zero of order p at ПЂ. To reproduce polynomials of degree 2 p в€’ 1, Theorem 7.21 proves that h ^ ( П‰ ) must have a zero of order 2 p at ПЂ. One can verify that such an interpolation function is the autocorrelation of a scaling function П† o. Deslauriers-Dubuc interpolation functions of degree 2 p в€’ 1 are compactly supported interpolation functions of minimal size that decompose polynomials of degree 2 p в€’ 1.

Matlab b spline fit