Hilbert transform by divide-and-conquer piecewise linear approximation

For many years it has been known that having a piecewise linear fitting of a function, one can obtain a good approximation of its Hilbert transform. There are few ways to determine the slopes of a broken line approximation, but the most popular approaches need to specify the breakpoints. In this paper we determine the breakpoints using a divide-and-conquer approach, then the Hilbert transform can be computed in two different ways. Simulations results provided here show that the piecewise linear approximation of function and the resulting Hilbert transform approximation are relatively accurate, and the complexity of implementation is not large.