Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials

Kirill A. Kopotun

doi:10.1007/bf01263061

Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials

Constructive Approximation ◽

10.1007/bf01263061 ◽

1994 ◽

Vol 10 (2) ◽

pp. 153-178 ◽

Cited By ~ 22

Author(s):

Kirill A. Kopotun

Keyword(s):

Algebraic Polynomials ◽

Approximation Of Functions ◽

Convex Approximation ◽

Uniform Estimates

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Some negative results for the interpolation monotone approximation of functions having a fractional derivative

Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics ◽

10.17721/1812-5409.2020/3.14 ◽

2020 ◽

pp. 122-127

Author(s):

T. O. Petrova ◽

I. P. Chulakov

Keyword(s):

Fractional Derivative ◽

Convex Function ◽

Nondecreasing Function ◽

Degree Of Approximation ◽

Approximation Of Functions ◽

Monotone Approximation ◽

Convex Approximation ◽

Negative Results ◽

Positive Functions ◽

Approximation Of A Function

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function $f є W^r [0,1]$ by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function $f є C^r [0,1] \cap \Delta^0$ where $\Delta^0$ is the set of 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],[8]. In [3],[4] is consider $r є , r > 2$. In [8] is consider $r є , r > 2$. It was proved that for monotone approximation estimates of the form (1) are fails for $r є , r > 2$. 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 that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6]). In [5] is consider $r є , r > 2$. In [6] is consider $r є , r > 2$. It was proved that for convex approximation estimates of the form (1) are fails for $r є , r > 2$. In this paper the question of approximation of function $f є W^r \cap \Delta^1, r є (3,4)$ by algebraic polynomial $p_n є \Pi_n \cap \Delta^1$ is consider. The main result of the work generalize the result of work [8] for $r є (3,4)$.

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