Generalization of Negative Results for Interpolation Convex Approximation of Functions Having a Fractional Derivative in Sobolev Space

• Дата: 2022
• Автори: Петрова, Тамара, Петрова, Ірина
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2022
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/274094
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function of Sobolev`s space by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function of r times continuously differentiable and positive functions on [0, 1]. Estimates of the form (1) for positive approximation are known ([1, 2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3, 4, 7]. In [3, 4] is consider r is natural and r not equal one. In [7] is consider r is real and r more two. It was proved that for monotone approximation estimates of the form (1) are fails for r is real and r more two. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in [8-10]. In [8] is consider r is natural and r not equal one. In [9] is consider r is real and r more two. It was proved that for convex approximation estimates of the form (1) are fails for r is real and r more two. In [10] the question of approximation of function of Sobolev`s space and convex by algebraic convex polynomial is consider, if the index of the Sobolev space is in the interval from three to four. It is proved that the estimate that generalizes (1) is false This paper investigates the issue of approximation of convex functions from the Sobolev space by convex algebraic polynomials for a real index of the Sobolev space from the interval from two to three. Similarly to the paper [10], a counterexample is built, which shows that the estimate that generalizes the estimate (1) is false. This paper is the generalization of results papers [9] and [11]. The main result is the analog of the theorem 2.3 in [11].