Comments on fast-filtering and a partition of unity

John C. Bancroft

Fast-filtering is a process that differentiates a filter operator a number of times and integrates the input data the same number of times. The differentiation diminishes the operator to a few delta functions that reduce convolution to the summation of few integrated samples. The filter operators are typically formed by repeated convolution of a boxcar with itself a number of times that will tend to a Gaussian shape when the number of convolutions becomes large. A finite number of these fast-filter operators will not form a partition of unity and are not suitable for decomposing a trace into short time windows to approximate stationary analysis. They do however indicate how operators can be created that will form a partition of unity and maintain the features of fast-filtering.