Thus Making the Recursion

X k=1 z 0k A0;k(z M)Mi01(z M): (41) This formula makes transparent exactly what happens in a recursive computation of the correlation functions. At every stage i, we have the Z-transform at points M 0i n of the appropriate function. In going to the (i + 1)st stage, the rst term Mi01(z M) ensures that values at the previous stage are retained at the even points, while the second term interpolates new values at the between points. This computation scheme is essentially the same as the wavelet-based lowpass/bandpass interpolation scheme in [5]. Let us examine Eqn. 38 more closely. Now Hl(z) has exactly 2N 01 points and the rst N 01 points can can be visualized graphically as M 01 times z }| { (42) where x stands for some non-zero number. Again for any of the cross-correlations Hl;m, the rst N 0 1 points in the sequence M1(z) can be visualized as in Eqn. 42. Now let us assume that this is true for all Mi(z), i = 1;2; : : : ; J. We then show that it is true for MJ+1(z), proving by induction that Mi(z) has for all i,