Upper Estimates of the angle best approximations of generalized Liouville-Weyl derivatives
Жалпыланған Лиувилл-Вейл туындыларының бұрыштық ең жақын жуықтауларының жоғарғы бағалаулары
Верхние оценки угловых наилучших приближений обобщенных производных Лиувилля-Вейля
Jetpisbayeva A.E. Jumabayeva A.A.
2021E.A.Buketov Karaganda State University Publish House
Bulletin of the Karaganda University. Mathematics Series
2021#103Issue 354 - 67 pp.
In this article we consider continuous functions f with period 2π and their approximation by trigonometric polynomials. This article is devoted to the study of estimates of the best angular approximations of generalized Liouville-Weyl derivatives by angular approximation of functions in the three-dimensional case. We consider generalized Liouville-Weyl derivatives instead of the classical mixed Weyl derivative. In choosing the issues to be considered, we followed the general approach that emerged after the work of the second author of this article. Our main goal is to prove analogs of the results of in the three-dimensional case. The concept of general monotonic sequences plays a key role in our study. Several well-known inequalities are indicated for the norms, best approximations of the r-th derivative with respect to the best approximations of the function f. The issues considered in this paper are related to the range of issues studied in the works of Bernstein. Later Stechkin and Konyushkov obtained an inequality for the best approximation f(r). Also, in the works of Potapov, using the angle approximation, some classes of functions are considered. In subsection 1 we give the necessary notation and useful lemmas. Estimates for the norms and best approximations of the generalized Liouville-Weyl derivative in the three-dimensional case are obtained.
best approximation by three-dimensional angle , Lebesgue space , Liouville-Weyl derivative , trigonometric polynomial
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L.N. Gumilyov Eurasian National University, Nur-Sultan, Kazakhstan
L.N. Gumilyov Eurasian National University
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