Preprint Article Version 1 This version is not peer-reviewed

Inequalities and Approximations for the Finite Hilbert Transform: A Survey of Recent Results

Version 1 : Received: 1 March 2018 / Approved: 2 March 2018 / Online: 2 March 2018 (05:11:07 CET)

How to cite: Dragomir, S.S. Inequalities and Approximations for the Finite Hilbert Transform: A Survey of Recent Results. Preprints 2018, 2018030017 (doi: 10.20944/preprints201803.0017.v1). Dragomir, S.S. Inequalities and Approximations for the Finite Hilbert Transform: A Survey of Recent Results. Preprints 2018, 2018030017 (doi: 10.20944/preprints201803.0017.v1).

Abstract

In this paper we survey some recent results due to the author concerning various inequalities and approximations for the finite Hilbert transform of a function belonging to several classes of functions, such as: Lipschitzian, monotonic, convex or with the derivative of bounded variation or absolutely continuous. More accurate estimates in the case that the higher order derivatives are absolutely continuous, are also provided. Some quadrature rules with error bounds are derived. They can be used in the numerical integration of the finite Hilbert transform and, due to the explicit form of the error bounds, enable the user to predict a priory the accuracy.

Subject Areas

Finite Hilbert Transform; Lipschitzian; Monotonic; Convex functions; Midpoint and Trapezoid inequalities; Ostrowski's inequality; Taylor's formula

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