Approximating The Finite Hilbert Transform Via Some Inequalities

Finite Hilbert transform has an important role in many fields, especially in commu-nication and signal processing. Finite part integral calculation is a hotspot of research, mainly using the numerical approximation method to construct the approximation function. It has developed many methods, and it has good approximation effect by using integral inequality. In this thesis some we use some new integral inequalities to approximate the finite Hilbert transform, get smaller approximation error. This thesis includes the following aspects:(1)Approximating the finite Hilbert transform via a companion of Ostrowski’s inequality for function of bounded variation. By use of bounded variation difference function with Ostrowski’s inequality, given the finite Hilbert transform some inequality approach. First based on interval [a, b], then of equidistant partition and further that non equidistant partition, the result has smaller error bounds.(2) Approximating the finite Hilbert transform via some companions of Ostrows-ki’s inequalities for absolutely continuous functions. We establish inequalities when functions belong to L1, LP and L∞ space respectively. First based on interval [a, b], then of equidistant partition. Compared with the results in the literature, there is a much smaller error bound.(3)Approximating the finite Hilbert transform via perturbed trapezoid type in-equalities for absolutely continuous functions. Study on the second order and third-order perturbed trapezoid inequalities. We establish inequalities when functions belong to L1, LP and L∞ space respectively. Giving the result on equidistant division of [a, b].Using companions of Ostrowski’s inequalities and perturbed trapezoid type in-equalities about finite Hilbert transform approximation, through specific function, ap-plication of the obtained results on the error estimation and the obtained approxima-tion inequality gives the corresponding function graphics and error analysis chart. The numerical experiments show that the approximation error is smaller.