Approximating The Finite Hilbert Transform Via Different Simpson Type Inequalities

The finite Hilbert transform plays an important role in Aerodynamics, commu-nication systems and digital signal processing, solving the differential equations and so on, so it has been concerned by the mathematical community. It is essentially a Cauchy principal value transformation of analytic functions within a limited range. In this thesis, we study the approximations of the finite Hilbert transform via different Simpson type inequalities, the main work includes the following aspects:(1) Approximating the finite Hilbert transform via the Simpson type inequalities for absolutely continuous functions. By establishing the Simpson type inequalities of the first derivative of a function is absolutely continuous on [a, b]、the second derivative of a function is absolutely continuous on [a, b], we get some approximations of the finite Hilbert transform.(2) Approximating the finite Hilbert transform via the Simpson type inequalities for absolutely continuous functions on an equidistant partition of the interval [a, b]. By establishing the Simpson type inequalities of the first derivative of a function is absolutely continuous on the equidistant division of [a, b] and the second derivative of a function is absolutely continuous on the equidistant division of [a, b], we get two approximations of the finite Hilbert transform.(3) Approximating the Finite Hilbert transform via Simpson type inequalities for functions of bounded variation. we have introduced a Simpson type inequality for functions of bounded variation and get a new approximation for the finite Hilbert transform. We further study a new quadrature rule on an equidistant partition and Arbitrary division of the interval [a, b] for the finite Hilbert transform via Simpson type inequalities.(4) With analytic functions, we find that the precision of the three errors we get here are higher than those of the existing literature.