Subdivision, total positivity and causality

We consider subdivision schemes which have good shape properties for design purposes because the resulting bases are totally positive. This happens when the symbol of the corresponding refinement equation has all its roots in the left half-plane, and includes the case of uniform B-splines. We then show that when a corresponding refinable function is used as the kernel of a smoothing operator, then this operator has a version of the `causality property´, i.e. the smoothed signal cannot increase its number of zero crossings as the scale of the kernel increases. This property is possessed by the Gaussian kernel, as has been much studied in connection with computer vision, and we examine how the above smoothing operators approach asymptotically the operator with Gaussian kernel.