The Joint Approximation (ψ, β) — integrals by Fejer’S sums in the metric Lp

• Дата: 2019
• Автори: Sorych, Viktor, Sorych, Nina
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2019
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/188991
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

We know, that any summable periodic function is answered its Fourier’s series. Therefore, it is naturally to approach it by trigonometric polynomials, that are the part’s sums of their series, they are named the Fourier’s sums. Sometimes the Fourier’s sums very slowly approach to it, sometimes they do not approach (the example of the continuous function with divergent Fourier’s series in some points was made Du Bois Reimond in 1876 y.). This fact induced mathematicians to search other sequences trigonometric polynomials, that they would gather to their generative function or that would uniformly gather to it on all space. Clear, that most successful in understanding of speed of convergence there is a sequence of its polynomials of the best approaching to the function. But, unfortunately, an operator of the best approaching is not linear. It complicates the construction of polynomials of the best approaching in large part, and, therefore, their use.If to examine the linear methods of summarization Fourier’s sums only, then matrix summarization the gives great class of such methods. One of these methods there is Fejer’s method, the method middle arithmetic of the first n Fourier’s sums.In this work the asymptotic equality are found at n → ∞ of upper bound of the value of the joint approximation by Fejer’s sums of the order n of functions, that have a derivative in sense of Stepanets, in the case of an achievement of the saturation, in the metric of space of integral at p degrees functions. The main member of an asymptotic decomposition is selected and the order of the residual member is shown also.