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# Polynomial operators for one-sided ${L}_{p}$-approximation to Riemann integrable functions

- Jose A Adell
^{1}Email author, - Jorge Bustamante
^{2}and - Jose M Quesada
^{3}

**2014**:494

https://doi.org/10.1186/1029-242X-2014-494

© Adell et al.; licensee Springer. 2014

**Received:**1 July 2014**Accepted:**25 November 2014**Published:**12 December 2014

## Abstract

We present some operators for one-sided approximation of Riemann integrable functions on $[0,1]$ by algebraic polynomials in ${L}_{p}$-spaces. The estimates for the error of approximation are given with an explicit constant.

## Keywords

- one-sided approximation
- operators for one-sided approximation
- Riemann integrable functions

## 1 Introduction

and let $C[0,1]$ be the set of all continuous functions $f:[0,1]\to \mathbb{R}$ with the sup norm ${\parallel f\parallel}_{\mathrm{\infty}}=sup\{|f(t)|:x\in [0,1]\}$. We simply write ${\parallel f\parallel}_{p}={\parallel f\parallel}_{p,[0,1]}$. Let $\mathcal{R}[0,1]$ be the set of all Riemann integrable functions on $[0,1]$ (recall that Riemann integrable functions on $[0,1]$ are bounded). As usual, we denote by ${W}_{p}^{1}[0,1]$ the space of all absolutely continuous functions $f:[0,1]\to \mathbb{R}$ such that ${f}^{\prime}\in {L}_{p}[0,1]$. We also denote by ${\mathbb{P}}_{n}$ the family of all algebraic polynomials of degree not greater than *n*.

*x*is defined by

This modulus is well defined whenever *g* is a bounded measurable function.

*C*such that, for any bounded measurable function $f:[0,1]\to \mathbb{R}$,

The analogous result for a trigonometric approximation was given in [2]. It is well known that ${lim}_{t\to 0+}\tau {(g,t)}_{1}=0$ if and only if $g\in \mathcal{R}[0,1]$ (see [3]). That is the reason why we only consider Riemann integrable functions.

In this paper, we present some sequences of polynomial operators for one-sided ${L}_{p}$-approximation which realize the rate of convergence given in (2). Our construction also provides a specific constant. We point out that a one-sided approximation cannot be realized with polynomial linear operators.

The existence of such sequences of polynomials ${P}_{n}$ and ${Q}_{n}$ satisfying (4) is well known (see, for example, [4]). Probably, the first construction of the optimal solution for (4) is due to Markoff or Stieltjes (*cf.* Szegö [[5], Section 3.411, p.50]).

and present upper estimates for the error $f-{A}_{n,h}(f)$ and ${B}_{n,h}(f)-f$ in ${L}_{p}[0,1]$ in terms of the average modulus of continuity.

In the last years, there has been interest in studying open problems related to one-sided approximations (see [4, 6–8] and [9]). We point out that other operators for the one-sided approximation have been constructed in [10, 11] and [12]. In particular, the operators presented in [12] yield the non-optimal rate $O(\tau {(f,1/\sqrt{n})}_{1})$ whereas the ones considered in [10, 11] give the optimal rate, but without an explicit constant.

The paper is organized as follows. In Section 2 we present some properties of the Steklov type functions (10) and (11). Finally, in Section 3 we consider an approximation by means of the operators defined in (12).

## 2 Properties of Steklov type functions

We start with the following auxiliary results.

**Proposition 1**

*If*$f\in \mathcal{R}[0,1]$, $h\in (0,1)$,

*and the functions*${L}_{h}(f)$

*and*${M}_{h}(f)$

*are defined by*(10)

*and*(11),

*respectively*,

*then the following assertions hold*.

- (i)
*The functions*${L}_{h}(f)$*and*${M}_{h}(f)$*are absolutely continuous*.*Moreover*,*if*${\mathrm{\Psi}}_{1}(x):={L}_{h}(f,x)$*and*${\mathrm{\Psi}}_{2}(x):={M}_{h}(f,x)$,*then*$\begin{array}{rl}{\mathrm{\Psi}}_{j}^{\prime}(x)=& \frac{1-h}{h}(f((1-h)x+h)-f((1-h)x))\\ +{(-1)}^{j}\frac{1-h}{h}(\omega (f,(1-h)x+h,h)-\omega (f,(1-h)x,h)),\phantom{\rule{1em}{0ex}}j=1,2.\end{array}$(13) - (ii)
*For each*$x\in [0,1]$,${L}_{h}(f,x)\le f(x)\le {M}_{h}(f,x).$(14) - (iii)
*For*$1\le p<\mathrm{\infty}$*one has*${L}_{h}{(f)}^{\prime},{M}_{h}{(f)}^{\prime}\in {L}_{p}[0,1]$$max\{{\parallel f-{L}_{h}(f)\parallel}_{p},{\parallel f-{M}_{h}(f)\parallel}_{p}\}\le \frac{2}{{(1-h)}^{1/p}}\tau {(f,h)}_{p},$(15)

*and*

*Proof*(i) Let $g\in {L}_{1}[0,1]$. Then the function

- (ii)Observe that$f(x)-{M}_{h}(f,x)=\frac{1}{h}{\int}_{0}^{h}(f(x)-f((1-h)x+s)-\omega (f,(1-h)x+s,h))\phantom{\rule{0.2em}{0ex}}ds\le 0,$

*ω*. Similarly, ${L}_{h}(f,x)\le f(x)$.

- (iii)We present a proof for a fixed $1<p<\mathrm{\infty}$ (the case $p=1$ follows analogously). As usual, take
*q*such that $1/p+1/q=1$. Using (14) and Hölder inequality, we obtain$\begin{array}{rcl}{\left(h{\parallel {M}_{h}(f)-f\parallel}_{p}\right)}^{p}& \le & {\left(h{\parallel {M}_{h}(f)-{L}_{h}(f)\parallel}_{p}\right)}^{p}\\ =& {2}^{p}{\int}_{0}^{1}{\left({\int}_{0}^{h}\omega (f,(1-h)x+s,h)\phantom{\rule{0.2em}{0ex}}ds\right)}^{p}\phantom{\rule{0.2em}{0ex}}dx\\ \le & {2}^{p}{h}^{p/q}{\int}_{0}^{1}{\int}_{0}^{h}{\omega}^{p}(f,(1-h)x+s,h)\phantom{\rule{0.2em}{0ex}}ds\phantom{\rule{0.2em}{0ex}}dx\\ =& \frac{{2}^{p}{h}^{p/q}}{1-h}{\int}_{0}^{h}{\int}_{s}^{1-h+s}{\omega}^{p}(f,y,h)\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{{2}^{p}{h}^{p/q}}{1-h}{\int}_{0}^{h}{\int}_{0}^{1}{\omega}^{p}(f,y,h)\phantom{\rule{0.2em}{0ex}}dy\phantom{\rule{0.2em}{0ex}}ds\\ =& \frac{{2}^{p}{h}^{1+p/q}}{1-h}{\tau}^{p}{(f,h)}_{p}=\frac{{2}^{p}{h}^{p}}{1-h}{\tau}^{p}{(f,h)}_{p}.\end{array}$

For ${\parallel f-{L}_{h}(f)\parallel}_{p}$ the proof follows analogously.

From (13), (18), and (19), we obtain (17). The proof is complete. □

## 3 Approximation of Riemann integrable functions

**Theorem 1**

*Fix*$p\in [1,\mathrm{\infty})$, $n\in \mathbb{N}$, $h\in (0,1)$,

*and*$f\in \mathcal{R}[0,1]$.

*Let*${A}_{n,h}(f)$

*and*${B}_{n,h}(f)$

*be as in*(12)

*and let*${\alpha}_{n}$

*be as in*(9).

*Then*${A}_{n,h}(f),{B}_{n,h}(f)\in {\mathbb{P}}_{n}$,

*and*

*Proof*Let ${L}_{h}(f)$ and ${M}_{h}(f)$ be as in (10) and (11), respectively. We know that ${A}_{n,h}(f),{B}_{n,h}(f)\in {\mathbb{P}}_{n}$. Moreover, from (7) and (14) we have

The estimate for ${\parallel f-{B}_{n,h}(f)\parallel}_{p}$ follows analogously. Finally, (21) follows immediately from (20). □

for $t\ne \pm \pi /(n+2)$ and ${K}_{n}(t)=(n+2)/2$ for $t=\pm \pi /(n+2)$.

*π*-periodic function such that

The following result was proved in [4].

**Proposition 2**

*Let*

*G*

*be given by*(3).

*For*$n\in \mathbb{N}$

*and*$x\in [-1,1]$,

*define*

*Then*${P}_{n},{Q}_{n}\in {\mathbb{P}}_{n}$,

*and*

From Theorem 1 and Proposition 2 we can state our main results.

**Theorem 2**

*Fix*$p\in [1,\mathrm{\infty})$.

*For*$n\in \mathbb{N}$,

*let*${P}_{n}$, ${Q}_{n}$

*be the sequences of polynomials constructed as in Proposition*2.

*For*$f\in \mathcal{R}[0,1]$

*and*$n\ge 2$,

*set*

*where*${A}_{n,h}$

*and*${B}_{n,h}$

*are given by*(12).

*Then*

*and*

*Proof*The first two assertions follow from Proposition 2 with $h=1/n$. So, in order to prove the theorem it remains to verify (24). Taking into account (9) and (23), we have ${\alpha}_{n}\le 4{\pi}^{2}/(n+2)$. Then, from (20) with $h=1/n$ and $n\ge 2$, we obtain

This completes the proof. □

Finally, from Theorem 2 we have immediately the following.

**Corollary 1**

*Fix*$p>1$

*and*$n\in \mathbb{N}$, $n\ge 2$.

*For any*$f\in \mathcal{R}[0,1]$

*we have*

*where* ${\tilde{E}}_{n}{(f)}_{p}$ *is the best one*-*sided approximation defined in* (1).

## Declarations

### Acknowledgements

The authors are partially supported by Research Project MTM2011-23998 and by Junta de Andalucía Research Group FQM268. This work was written during the stay of JAA and JB at the Department of Mathematics of the University of Jaén in May 2014.

## Authors’ Affiliations

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