Jackson’s rational singular integral on the cut

Yevgeniy A Rovba,Pavel G Potsejko

doi:10.29235/1561-8323-2019-63-4-398-407

Yevgeniy A Rovba, Pavel G Potsejko

Open Access

https://doi.org/10.29235/1561-8323-2019-63-4-398-407

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Abstract

The introduction presents the main results of previously known papers on Jackson’s singular integral in polynomial and rational cases. Next, we introduce Jackson’s singular integral on the interval [–1, 1] with the kernel obtained by one system of rational Chebyshev–Markov fractions and establish its basic approximative properties: a theorem on uniform convergence of a sequence of Jackson’s singular integrals for an even function is obtained, and conditions are specified that the parameter must satisfy in order for uniform convergence to take place; the approximative properties of sequences of Jackson’s singular integrals on classes of functions satisfying on the interval [–1, 1] the condition of Lipschitz class with constant M. are investigated. The obtained estimates are asymptotically exact asn→ ∞; an estimate of deviation of Jackson’s rational singular integral from the function |x|s, 0 <s< 2 depending on the position of the point on the segment, a uniform estimate of the deviation on the segment [–1, 1] and its asymptotics are found. The optimal value of the parameter is obtained, for which the deviation error of the studied approximation apparatus from the function |x|s, 0 <s< 2 on the interval [–1, 1] has the highest rate of zero; the order of approximation of the function |x| on the interval [–1, 1] byJackson’s considered singular integral is found. It is shown that with a special choice of the parameter, the velocity of the approximation error tending to zero is higher in comparison with the polynomial case. All results of this article are new. The work is both theoretical and applied. It is possible to apply the results to solve specific problems of computational mathematics and to read special courses at mathematical faculties.

Highlights

The introduction presents the main results of previously known papers on Jackson’s singular integral in polynomial and rational cases
We introduce Jackson’s singular integral on the interval [–1, 1] with the kernel obtained by one system of rational Chebyshev–Markov fractions and establish its basic approximative properties: a theorem on uniform convergence of a sequence of Jackson’s singular integrals for an even function is obtained, and conditions are specified that the parameter must satisfy in order for uniform convergence to take place; the approximative properties of sequences of Jackson’s singular integrals on classes of functions satisfying on the interval [–1, 1] the condition of Lipschitz class with constant M. are investigated
Н. Бернштеин // Известия Академии наук СССР

Summary

Имеет место

X ∈[-1,1], x =cosu, где ω f – модуль непрерывности функции f на отрезке [–1, 1], l(u) из (2). Для оценки погрешности приближения функции | x |s ,0 < s < 2, на отрезке [–1, 1] сингулярным интегралом Джексона U 4n (| x |s , x) справедливо соотношение. Для приближения функции | x |s , 0 < s < 2, на отрезке [–1, 1] сингулярным интегралом Джексона U 4n (| x |s , x) имеет место равномерная оценка где = I1(α, n). Для мажоранты равномерной оценки приближений функции | x |s , s ∈ (0, 2), на отрезке [–1, 1] сингулярным интегралом Джексона (3) при n → ∞ имеет место асимптотическая формула β s. Для равномерной оценки приближений функции | x |s , s ∈ (0, 2), на отрезке [–1, 1] сингулярным интегралом Джексона (3) справедливо асимптотическое соотношение.

При этом оптимальным значением параметра для достижения оценки справа будет β*

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