Why the causal wavelet can be a divisor

 The danger of polynomial division is that the coefficients will diverge like this, $1/(1-4Z)= 1 +4Z +16Z^2 +64Z^3 +256Z^4 +1024Z^5 + \cdots$.The polynomial 1-4Z is an example of one that cannot be used in polynomial division. Our factoring of operators like the Laplacian gives us one factor that works with polynomial division in the forward time direction and a reversed one for the backward time direction.

Next we see why the causal wavelet B(Z), of the Kolmogoroff theory can be a divisor, namely, why the inverse does not diverge. We have our wavelet in the form B(Z)= e^C(Z). Consider another wavelet A(Z) = e^-C(Z), constructed analogously. By the same reasoning, a_t is also causal. Since A(Z)B(Z)=1, we have found a causal, inverse wavelet. This means that polynomial division with B(Z) should not give divergent coefficients. This property of the b_t wavelet is often called minimum-phase, for reasons found in my book PVI. Other notable properties of a minimum-phase wavelet are: (1) Poles and zeros are outside the unit circle in the |Z|-plane; (2) Equivalent to transmitted waves in a layered medium where reflection coefficients are weaker than unity; (3) Phase at minus Nyquist equals that at plus Nyquist. (Observe this is not true for the pure delay operator $Z=e^{i\omega}$.)