Wavelet and Scaling Function Moments 2.1 2003/02/14 2003/05/09 Ivan Selesnick selesi@taco.poly.edu Charlet Reedstrom charlet@rice.edu Ivan Selesnick selesi@taco.poly.edu moment scaling function wavelets Define m k t t k φ t μ k n n k h n Then m 1 t t φ t μ 1 n n h n It turns out we can relate m 1 to the moments of h n by using the dilation equation. m 1 t t φ t t t 2 n h n φ 2 t n 2 n h n t t φ 2 t n 2 n h n 1 2 α α 2 n 2 φ α 2 4 n h n α α φ α 2 4 n h n α n φ α 2 4 n h n m 1 2 4 n n h n m 0 2 4 m 1 n h n 2 4 m 0 n n h n 2 4 m 1 μ 0 2 4 m 0 μ 1 So we have m 1 2 4 m 1 μ 0 2 4 m 0 μ 1 Solving for m 1 gives m 1 2 m 0 μ 1 4 2 μ 0 A valid scaling filter h n must have μ 0 2 . Using this, we get m 1 m 0 μ 1 2 With the normalization m 0 t φ t 1 , we get m 1 μ 1 2