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.

Two Sharp Perturbed Midpoint Inequalities

2014 ◽  
Vol 472 ◽  
pp. 527-531
Author(s):  
Yan Xia Shi ◽  
Yu Min Tao ◽  
Yu Pan
Keyword(s):  
Numerical Integration ◽  
Error Bounds ◽  
Kernel Functions ◽  
Quadrature Rules

In this study, two new sharp perturbed midpoint inequalities are proved by establishing proper kernel functions. These results enlarge applicability of the corresponding quadrature rules with respect to the obtained error bounds. Applications in numerical integration are also given.

scholarly journals Perturbations of an Ostrowski type inequality and applications

2002 ◽  
Vol 32 (8) ◽  
pp. 491-500 ◽  
Author(s):  
Nenad Ujević
Keyword(s):  
Numerical Integration ◽  
Error Bounds ◽  
Type Inequality ◽  
Quadrature Rules ◽  
Ostrowski Type Inequality ◽  
Better Than

Two perturbations of an Ostrowski type inequality are established. New error bounds for the mid-point, trapezoid, and Simpson quadrature rules are derived. These error bounds can be much better than some recently obtained bounds. Applications in numerical integration are also given.

scholarly journals Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Elena E. Berdysheva ◽  
Nira Dyn ◽  
Elza Farkhi ◽  
Alona Mokhov
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Error Bounds ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Dirichlet Kernel ◽  
Partial Sums ◽  
Functions Of Bounded Variation ◽  
Moduli Of Continuity ◽  
Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

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