Application of Bernstein polynomials to suppress the Gibbs effect (literature review) 

Il'ya V. Boykov, Doctor of physical and mathematical sciences, professor, head of the sub-department of higher and applied mathematics, Penza State University (40 Krasnaya street, Penza, Russia), boikov@pnzgu.ru
Georgiy Yu. Salimov, Postgraduate student, Penza State University (40 Krasnaya street, Penza, Russia), salimovgorik@mail.ru 

519.65 

10.21685/2072-3040-2021-4-7 

Background. Despite Gibbs effect (phenomenon) was discovered almost 170 years ago, the amount of works devoted to its research and the construction of methods of its suppression has not weakened until recently. This is due to the fact that the Gibbs effect has a negative impact on the study of many wave processes in hydrodynamics, electrodynamics, microwave technology, and computational mathematics. Therefore, the construction of new methods for suppressing the Gibbs effect is an issue of the day. In addition, several computational schemes for solving the Gibbs problem in one particular formulation are proposed. Materials and methods. Methods of approximation theory were used in the construction of computational schemes. In particular, the properties of Bernstein polynomials were used in the approximation of integer functions. Results. The review of works devoted to the study of the Gibbs effect and methods of constructing filters that suppress this effect is presented. The review includes: historical background on the study of the Gibbs effect; various methods of suppressing the Gibbs effect; methods of building filters; descriptions of the manifestation of the Gibbs effect in technology. Conclusions. The possibility of applying Bernstein polynomials to the solution of the Gibbs problem in
the case of analytic nonperiodic functions given by values in equally spaced nodes is demonstrated. These results can be used in solving the problem of suppressing the Gibbs effect in other formulations. 

Gibbs effect, filters, Bernstein polynomial 

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