Approximating the Finite Hilbert Transform for Absolutely Continuous Mappings and Applications in Numerical Integration

2018 ◽  
Vol 28 (4) ◽  
Author(s):  
Fuat Usta
Keyword(s):  
Numerical Integration ◽  
Hilbert Transform ◽  
Absolutely Continuous ◽  
Finite Hilbert Transform ◽  
Continuous Mappings

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

2018 ◽  
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Numerical Integration ◽  
Explicit Form ◽  
Bounded Variation ◽  
Hilbert Transform ◽  
Error Bounds ◽  
Higher Order ◽  
Quadrature Rules ◽  
Absolutely Continuous ◽  
Finite Hilbert Transform

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.

scholarly journals Approximating the finite Hilbert transform via trapezoid type inequalities

2002 ◽  
Vol 43 (10-11) ◽  
pp. 1359-1369 ◽  
Author(s):  
N.M. Dragomir ◽  
S.S. Dragomir ◽  
P. Farrell
Keyword(s):  
Hilbert Transform ◽  
Finite Hilbert Transform
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