Abstract
The approximative properties of the Valle Poussin means of the Fourier series by the system of the Chebyshev – Markov rational fractions in the approximation of the function |x|s, 0 < s < 2 are investigated. The introduction presents the main results of the previously known works on the Vallee Poussin means in the polynomial and rational cases, as well as on the known literature data on the approximations of functions with power singularity. The Valle Poussin means on the interval [–1,1] as a method of summing the Fourier series by one system of the Chebyshev – Markov rational fractions are introduced. In the main section of the article, a integral representation for the error of approximations by the rational Valle Poussin means of the function |x|s, 0 < s < 2, on the segment [–1,1], an estimate of deviations of the Valle Poussin means from the function |x|s, 0 < s < 2, depending on the position of the point on the segment, a uniform estimate of deviations on the segment [–1,1] and its asymptotic expression are found. The optimal value of the parameter is obtained, at which the deviation error of the Valle Poussin means from the function |x|s, 0 < s <2, on the interval [–1,1] has the highest velocity of zero. As a consequence of the obtained results, the problem of approximation of the function |x|s, s > 0, by the Valle Poussin means of the Fourier series by the system of the Chebyshev first-kind polynomials is studied in detail. The pointwise estimation of approximation and asymptotic estimation are established.The work is both theoretical and applied. Its results can be used to read special courses at mathematical faculties and to solve specific problems of computational mathematics.
Highlights
The approximative properties of the Valle Poussin means of the Fourier series by the system
The Valle Poussin means on the interval
a integral representation for the error of approximations by the rational Valle Poussin means of the function |x|s
Summary
Для средних Валле Пуссена (4) имеет место представление π(n. Переходя к полиномиальному случаю, функция V4n( f,x) представляет собой классические средние Валле Пуссена рядов Фурье – Чебышева при условии четности функции f. Для приближений функции | x |s , 0 < s < 2, на отрезке [–1,1] средними Валле Пуссена (3) справедливо соотношение. Что V4n (1, x) ≡ 1, из (5) получим ( x, ). Выполнив в интеграле справа замены ζ = eiv , ξ = eiu и приняв во внимание (6), находим, что ε 4n (x,α) =. Интеграл справа разобьем на четыре интеграла так, что ε x, α) ξ2−s πi(n + 1)λ(u). Что для интеграла (12) точка ζ = ξ является нулем второго порядка как числителя, так и знаменателя подынтегрального выражения, и, следовательно, не будет особой
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