# On the Bernstein’s constant in convex approximation

Sorin G. Gal  Department of Mathematics and Computer Science, University of
Oradea, 410087 Oradea, Romania<EMAIL_ADDRESS>
###### Abstract.

Denoting by $E_{n}^{(+2)}(f)$ the best uniform approximation of $f$ by convex
polynomials of degree $\leq n$, there is an open question if there exists the
limit $\lim_{n\to\infty}n^{\lambda}E_{n}^{(+2)}(|x|^{\lambda})$ for
$\lambda\geq 1$.

2010 Mathematics Subject Classification. 41.

##  1\. Introduction

A famous result of Bernstein [1], [2] states that for any $\lambda>0$,
$\lambda$ not even integer, there exists finite the limit
$\lim_{n\to\infty}n^{\lambda}E_{n}(|x|^{\lambda})>0$, where

$E_{n}(|x|^{\lambda})=\inf\\{\max\\{|P(x)-|x|^{\lambda}|;x\in[-1,1]\\};P\in{\mathcal{P}}_{n}(\mathbb{R})\\},$

and ${\mathcal{P}}_{n}(\mathbb{R})$ denotes the set of all real polynomials
(that is with real coefficients) of degree $\leq n$.

It is worth noting that the exact values of the above limits are not known.
Details and generalizations of these results, can be found in the papers
[4]-[6], [10] and in the book [7].

In this note we consider a similar problem in the case of the best
approximation by convex polynomials, which was first mentioned in [3], p. 325.

##  2\. Convex Approximation

Taking into account that $|x|^{\lambda}$ is convex for $\lambda\geq 1$,
denoting

$E_{n}^{(+2)}(f)=\inf\\{\|f-P\|_{\infty};P\in{\mathcal{P}}_{n}(\mathbb{R}),P^{\prime\prime}(x)\geq
0,\forall x\in[-1,1]\\},$

it is natural to consider the following open question.

Open Question 1. There exists finite the limit
$\lim_{n\to\infty}n^{\lambda}E_{n}^{(+2)}(|x|^{\lambda})$ for $\lambda\geq 1$
?

Note that since by [8], for $f$ convex on $[-1,1]$ and $\lambda>0$ we have

$E_{n}(f)={\mathcal{O}}(n^{-\lambda})\quad\mbox{ iff
}E_{n}^{(+2)}(f)={\mathcal{O}}(n^{-\lambda}),\quad n\to\infty,$

it easily follows that the sequence
$(n^{\lambda}E_{n}^{(+2)}(|x|^{\lambda}))_{n\in\mathbb{N}}$ for $\lambda\geq
1$ is bounded.

More general, we can consider the following.

Open Question 2. If $f$ is convex on $\mathbb{R}$ and the sequence
$(\lambda_{n})_{n}$ satisfies the conditions in [7], p. 3, then it is valid
that we have

$\lim_{n\to\infty}E_{n}^{(+2)}\left(f;L_{\infty}\left[-\frac{n(1-\lambda_{n})}{\sigma},\frac{n(1-\lambda_{n})}{\sigma}\right]\right)<\infty(\mbox{
or }\leq A_{\sigma}^{(+2)}(f,L_{\infty}(\mathbb{R}))<\infty)\quad\quad?$

Here
$A_{\sigma}^{(+2)}(f,L_{\infty}(\mathbb{R})=\inf\\{\|f-g\|_{L_{\infty}(\mathbb{R})};g\in
B_{\sigma},g\mbox{ convex on }\mathbb{R}\\}$ and $B_{\sigma}$ denotes the
class of all entire functions of exponential type $\sigma$.

This open question is partially supported by the fact that by [9], for a
convex function $f$ we have $E_{n}^{(+2)}(f)_{L_{p}}\leq
E_{n-2}(f^{\prime\prime})_{L_{p}}$, for all $n\geq 2$ and $p=\infty$ and this
estimate cannot be improved.

## References

  * [1] S. N. Bernstein, Sur la meilleure approximation de $|x|$ par les polynomes des degrés donnés, Acta Mathematica, 37(1914), 1-57.
  * [2] S. N. Bernstein, On the best approximation of $|x|^{p}$ by polynomials of very high degree, Izv. Math. Nauk, SSSR, 1938, 169-180.
  * [3] S. G. Gal, Approximation by Complex Bernstein and Convolution Type Operators, World Scientific Publ., New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, 2009.
  * [4] M. I. Ganzburg, Limit theorems for the best polynomial approximations in the $L_{\infty}$ metric, Ukrainian Math. J., 43(1991), 299-305.
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  * [6] M. I. Ganzburg, The Bernstein constant and polynomial interpolation at the Chebyshev nodes, J. Approx. Theory, 119(2002), 193-213.
  * [7] M. I. Ganzburg, Limit Theorems of Polynomial Approximation with Exponential Weights, Memoirs of the American Mathematical Society, vol. 192, No. 897, Providence, Rhode Island, 2008.
  * [8] K. A. Kopotun, D. Leviatan and I. A. Shevchuk, Coconvex approximation in the uniform norm-the final frontier, Acta Math. Hungarica, 110(2006), No. 1-2, 117-151.
  * [9] D. Leviatan and I. A. Shevchuk, Counterexamples in convex and higher order constrained approximation, East J. Approx., 1(1995), 391-398.
  * [10] D. S. Lubinsky, On the Bernstein constants of polynomial approximation, Constr. Approx., 25(2007), 303-366.