On Hilbert Transform of Hypersingular Equation and Caldron-Zygmund operators Applications

Abstract

The information about the Hilbert transform is often scattered in books about signal processing. Their authors frequently use mathematical formulas without explaining them thoroughly to the reader. The purpose of this research is to make a more stringent presentation of the Hilbert transform but still with the signal processing application in mind. Everybody working in the field of singular integrals and integral equations will know that during the last few decades an entirely new mathematical field of Cauchy principal value integrals and hypersingular integral has developed. Since this is a recent mathematical development, it is not always easy for readers including academics, engineers and researchers, to get a grap on. Hilbert transforms deal with Calderon-Zygmund operators and the theory of Calderon-Zygmund operators in such a way that any body will be able to repeat the schedule use of Hilbert transforms.