Polynomial operators for one-sided $L_p$-approximation to Riemann integrable functions

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.

1 Introduction

Let $L_p[a,b]$ ($1\le p<\mathrm{\infty }$) be the space of all real-valued Lebesgue measurable functions, $f:[a,b]\to \mathbb{R}$, such that

$${\parallel f\parallel }_{p,[a,b]}={\left({\int }_a^b\right|f(t){|}^{p}dt)}^{1/p}<\mathrm{\infty }$$

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.

The local modulus of continuity of a function $g:[0,1]\to \mathbb{R}$ at a point x is defined by

$$\omega (g,x,t)=sup\left\{\right|g(v)-g(w)|:v,w\in [x-t,x+t]\cap [0,1]\},t\ge 0.$$

For $p\ge 1$, the average modulus of continuity is defined by

$$\tau {(g,t)}_p={\parallel \omega (g,\cdot ,t)\parallel }_p.$$

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

For a bounded function $f\in L_p[0,1]$ and $n\in \mathbb{N}$, the best one-sided approximation is defined by

$${\tilde{E}}_n{(f)}_p=inf\{{\parallel P-Q\parallel }_p:P,Q\in {\mathbb{P}}_n,Q(x)\le f(x)\le P(x),x\in [0,1]\}.$$

(1)

It was shown in [1], with ${\mathrm{\Delta }}_n=n^{-2}+n^{-1}\sqrt{1-t^2}$, that there exists a constant C such that, for any bounded measurable function $f:[0,1]\to \mathbb{R}$,

$${\tilde{E}}_n{(f)}_p\le C\tau {(f,{\mathrm{\Delta }}_n)}_p.$$

(2)

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.

Let us consider the step function

$$G(t)=\{\begin{array}{ll}0,& \text{if }-1\le t\le 0,\\ 1,& \text{if }0<t\le 1,\end{array}$$

(3)

and fix two sequences of polynomials $\{P_n\}$ and $\{Q_n\}$ ($P_n,Q_n\in {\mathbb{P}}_n$) such that

$$P_n(t)\le G(t)\le Q_n(t),t\in [-1,1]$$

and

$${\parallel Q_n-P_n\parallel }_{1,[-1,1]}\to 0,\text{as }n\to \mathrm{\infty }.$$

(4)

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]).

Fix $p\in [1,\mathrm{\infty })$. In [4] we constructed a sequence of polynomial operators as follows. For $n\in \mathbb{N}$, $f\in W_p^1[0,1]$, and $x\in [0,1]$, define

$${\lambda }_n(f,x)=f(0)+{\int }_0^1{P}_n(t-x){\left(f^{\prime }\right)}_{+}(t)dt-{\int }_0^1{Q}_n(t-x){\left(f^{\prime }\right)}_{-}(t)dt$$

(5)

and

$${\mathrm{\Lambda }}_n(f,x)=f(0)+{\int }_0^1{Q}_n(t-x){\left(f^{\prime }\right)}_{+}(t)dt-{\int }_0^1{P}_n(t-x){\left(f^{\prime }\right)}_{-}(t)dt,$$

(6)

where, as usual,

$$g_{+}(x)=max\{0,g(x)\}\text{and}{g}_{-}(x)=max\{0,-g(x)\}.$$

Also in [4], it is proved that ${\lambda }_n(f),{\mathrm{\Lambda }}_n(f)\in {\mathbb{P}}_n$,

$${\lambda }_n(f,x)\le f(x)\le {\mathrm{\Lambda }}_n(f,x),x\in [0,1]$$

(7)

and

$$max\{{\parallel f-{\lambda }_n(f)\parallel }_p,{\parallel f-{\mathrm{\Lambda }}_n(f)\parallel }_p\}\le {\alpha }_n{\parallel f^{\prime }\parallel }_p,$$

(8)

where

$${\alpha }_n={\parallel Q_n-P_n\parallel }_{1,[-1,1]}.$$

(9)

For a function $f\in \mathcal{R}[0,1]$, $h\in (0,1)$ and $x\in [0,1]$, set

$$L_h(f,x)={\int }_0^1[f((1-h)x+hs)-\omega (f,(1-h)x+hs,h)]ds$$

(10)

and

$$M_h(f,x)={\int }_0^1[f((1-h)x+hs)+\omega (f,(1-h)x+hs,h)]ds.$$

(11)

It turns out (see Section 2) that $L_h(f),M_h(f)\in W_p^1[0,1]$, for $p\ge 1$, and therefore we can define

$$A_{n,h}(f,x)={\lambda }_n(L_h(f),x)\text{and}{B}_{n,h}(f,x)={\mathrm{\Lambda }}_n(M_h(f),x),$$

(12)

where ${\lambda }_n$ and ${\mathrm{\Lambda }}_n$ are given by (5) and (6), respectively. We will prove that

$$A_{n,h}(f,x)\le f(x)\le B_{n,h}(f,x),x\in [0,1],$$

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

  $$\begin{array}{rl}{\mathrm{\Psi }}_j^{\prime }(x)=& \frac{1-h}{h}(f((1-h)x+h)-f((1-h)x))\\ +{(-1)}^j\frac{1-h}{h}(\omega (f,(1-h)x+h,h)-\omega (f,(1-h)x,h)),j=1,2.\end{array}$$

  (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)

$${\parallel M_h(f)-L_h(f)\parallel }_p\le \frac{2}{{(1-h)}^{1/p}}\tau {(f,h)}_p$$

(16)

and

$$max\{{\parallel L_h^{\prime }(f)\parallel }_p,{\parallel M_h^{\prime }(f)\parallel }_p\}\le \frac{3}{h}\tau {(f,h)}_p.$$

(17)

Proof (i) Let $g\in L_1[0,1]$. Then the function

$$H(x)={\int }_0^{1}g((1-h)x+hs)ds,$$

is absolutely continuous with Radon-Nikodym derivative

$$H^{\prime }(x)=\frac{1-h}{h}(g((1-h)x+h)-g((1-h)x)).$$

This, together with (10) and (11), shows (13).

• (ii) Observe that

  $$f(x)-M_h(f,x)=\frac{1}{h}{\int }_0^h(f(x)-f((1-h)x+s)-\omega (f,(1-h)x+s,h))ds\le 0,$$

as follows from the definition of 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

  $$\begin{array}{rcl}{\left(h{\parallel M_h(f)-f\parallel }_p\right)}^p& \le & {\left(h{\parallel M_h(f)-L_h(f)\parallel }_p\right)}^p\\ =& 2^p{\int }_0^1{\left({\int }_0^h\omega (f,(1-h)x+s,h)ds\right)}^{p}dx\\ \le & 2^p{h}^{p/q}{\int }_0^1{\int }_0^h{\omega }^p(f,(1-h)x+s,h)dsdx\\ =& \frac{2^p{h}^{p/q}}{1-h}{\int }_0^h{\int }_s^{1-h+s}{\omega }^p(f,y,h)dyds\\ \le & \frac{2^p{h}^{p/q}}{1-h}{\int }_0^h{\int }_0^1{\omega }^p(f,y,h)dyds\\ =& \frac{2^p{h}^{1+p/q}}{1-h}{\tau }^p{(f,h)}_p=\frac{2^p{h}^p}{1-h}{\tau }^p{(f,h)}_p.\end{array}$$

For ${\parallel f-L_h(f)\parallel }_p$ the proof follows analogously.

Finally, in order to estimate ${\parallel L_h{(f)}^{\prime }\parallel }_p$ and ${\parallel M_h{(f)}^{\prime }\parallel }_p$, we use the representation of the derivative given in (13). Recall that the usual modulus of continuity for $f\in L_p[0,1]$ is defined as follows:

$$\omega {(f,h)}_p=\underset{0<t\le h}{sup}{\left({\int }_0^{1-t}\right|f(x+t)-f(x){|}^{p}dx)}^{1/p}.$$

It can be proved (see the proof of Lemma 4 in [2]) that, for any $f\in \mathcal{R}[0,1]$,

$$\omega {(f,h)}_p\le \tau {(f,h)}_p.$$

Therefore

$$\begin{array}{c}{\left({\int }_0^1\right|f((1-h)x+h)-f((1-h)x){|}^{p}dx)}^{1/p}\hfill \\ ={\left(\frac{1}{1-h}{\int }_0^{1-h}\right|f(y+h)-f(y){|}^{p}dy)}^{1/p}\le \frac{\omega {(f,h)}_p}{{(1-h)}^{1/p}}\le \frac{\tau {(f,h)}_p}{{(1-h)}^{1/p}}.\hfill \end{array}$$

(18)

On the other hand

$$\begin{array}{c}{\left({\int }_0^1\right|\omega (f,(1-h)x+h,h)-\omega (f,(1-h)x,h){|}^{p}dx)}^{1/p}\hfill \\ \le {\left({\int }_0^1{\omega }^p(f,(1-h)x+h,h)dx\right)}^{1/p}+{\left({\int }_0^1{\omega }^p(f,(1-h)x,h)dx\right)}^{1/p}\hfill \\ \le \frac{1}{{(1-h)}^{1/p}}[{({\int }_h^1{\omega }^p(f,y,h)dy)}^{1/p}+{({\int }_0^{1-h}{\omega }^p(f,y,h)dy)}^{1/p}]\hfill \\ \le \frac{2}{{(1-h)}^{1/p}}\tau {(f,h)}_p.\hfill \end{array}$$

(19)

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 $A_{n,h}(f),B_{n,h}(f)\in {\mathbb{P}}_n$,

$$\begin{array}{c}{A}_{n,h}(f,x)\le f(x)\le B_{n,h}(f,x),x\in [0,1],\hfill \\ max\{{\parallel f-A_{n,h}(f)\parallel }_p,{\parallel f-B_{n,h}(f)\parallel }_p\}\le (\frac{2}{1-h}+\frac{3{\alpha }_n}{h})\tau {(f,h)}_p\hfill \end{array}$$

(20)

and

$${\parallel B_{n,h}(f)-A_{n,h}(f)\parallel }_p\le (\frac{4}{1-h}+\frac{6{\alpha }_n}{h})\tau {(f,h)}_p.$$

(21)

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

$$A_{n,h}(f)={\lambda }_n(L_h(f))\le L_h(f)\le f\le M_h(f)\le {\mathrm{\Lambda }}_n(M_h(f))=B_{n,h}(f).$$

On the other hand, from (8), (15), and (17) one has

$$\begin{array}{rcl}{\parallel f-A_{n,h}(f)\parallel }_p& \le & {\parallel f-L_h(f)\parallel }_p+{\parallel L_h(f)-A_{n,h}(f)\parallel }_p\\ \le & \frac{2}{{(1-h)}^{1/p}}\tau {(f,h)}_p+{\alpha }_n{\parallel {(L_{h}f)}^{\prime }\parallel }_p\\ \le & (\frac{2}{{(1-h)}^{1/p}}+\frac{3{\alpha }_n}{h})\tau {(f,h)}_p\\ \le & (\frac{2}{1-h}+\frac{3{\alpha }_n}{h})\tau {(f,h)}_p.\end{array}$$

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

For $n\in \mathbb{N}$ the Fejér-Korovkin kernel is defined by

$$K_n(t)=\frac{2{sin}^2(\pi /(n+2))}{n+2}{\left(\frac{cos((n+2)t/2)}{cos(\pi /(n+2))-cost}\right)}^2$$

for $t\ne \pm \pi /(n+2)$ and $K_n(t)=(n+2)/2$ for $t=\pm \pi /(n+2)$.

Let $F:\mathbb{R}\to \mathbb{R}$ be the 2π-periodic function such that

$$F(x)=\{\begin{array}{ll}1,& \text{if }x\in [-\pi /2,\pi /2],\\ 0,& \text{if }x\in [-\pi ,\pi ]\setminus [-\pi /2,\pi /2],\end{array}$$

(22)

and, for $x,u,t\in [-\pi ,\pi ]$, set

$$U(F,x,t,u)=\omega (F,x+t,|u|)+\omega (F,x+u,|t|).$$

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

$$T_n^{-}(x)=\frac{1}{4{\pi }^2}{\int }_{-\pi }^{\pi }({\int }_{-\pi }^{\pi }[F(x+t)-U(F,x,t,u)]K_n(t)dt)K_n(u)du$$

and

$$T_n^{+}(x)=\frac{1}{4{\pi }^2}{\int }_{-\pi }^{\pi }({\int }_{-\pi }^{\pi }[F(x+t)+U(F,x,t,u)]K_n(t)dt)K_n(u)du.$$

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

$$P_n(x)=T_n^{-}(arccosx)\mathit{\text{and}}{Q}_n(x)=T_n^{+}(arccosx).$$

Then $P_n,Q_n\in {\mathbb{P}}_n$,

$$P_n(x)\le G(x)\le Q_n(x),x\in [-1,1]$$

and

$${\parallel Q_n-P_n\parallel }_{1,[-1,1]}\le \frac{4{\pi }^2}{n+2}.$$

(23)

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

$${\mathcal{A}}_n(f)=A_{n,\frac{1}{n}}(f),{\mathcal{B}}_n(f)=B_{n,\frac{1}{n}}(f),$$

where $A_{n,h}$ and $B_{n,h}$ are given by (12). Then

$$\begin{array}{c}{\mathcal{A}}_n(f),{\mathcal{B}}_n(f)\in {\mathbb{P}}_n,\hfill \\ {\mathcal{A}}_n(f,x)\le f(x)\le {\mathcal{B}}_n(f,x),x\in [0,1],\hfill \\ max\{{\parallel f-{\mathcal{A}}_n(f)\parallel }_p,{\parallel f-{\mathcal{B}}_n(f)\parallel }_p\}\le 2(1+6{\pi }^2)\tau {(f,\frac{1}{n})}_p\hfill \end{array}$$

(24)

and

$${\parallel {\mathcal{B}}_n(f)-{\mathcal{A}}_n(f)\parallel }_p\le 4(1+6{\pi }^2)\tau {(f,\frac{1}{n})}_p.$$

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

$$\begin{array}{rcl}max\{{\parallel f-{\mathcal{A}}_n(f)\parallel }_p,{\parallel f-{\mathcal{B}}_n(f)\parallel }_p\}& \le & (\frac{2n}{n-1}+\frac{12{\pi }^{2}n}{n+2})\tau {(f,\frac{1}{n})}_p\\ \le & 2(1+6{\pi }^2)\tau {(f,\frac{1}{n})}_p.\end{array}$$

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

$${\tilde{E}}_n{(f)}_p\le 4(1+6{\pi }^2)\tau {(f,\frac{1}{n})}_p,$$

where ${\tilde{E}}_n{(f)}_p$ is the best one-sided approximation defined in (1).