Weighted quadrature rules via Grüss type inequalities for weighted L^p spaces

Abstract

New estimates of Grüss type for weighted Čebyšev functional in weighted L^p spaces are presented by using the Sonin’s identity. These results are applied to obtain new Ostrowski type inequalities and further to weighted numerical rules for functions whose derivative belongs to weighted L^p spaces.

Introduction

For two functions $f,g:[a,b]\to \mathbb{R},$ such that $f,g,fg\in L_{[a,b]}^1$ the Čebyšev functional is given by

$$T(f,g)=\frac{1}{b-a}{\int }_a^{b}f\left(x\right)g\left(x\right)dx-\frac{1}{{(b-a)}^2}\left({\int }_a^{b}f\left(x\right)dx\right)\left({\int }_a^{b}g\left(x\right)dx\right).$$

Here, the symbol $L_{[a,b]}^p$ (1 ≤ p < ∞) denotes the space of p-power integrable functions on the interval [a, b] equipped with the norm

$${\parallel f\parallel }_p={\left({\int }_a^b{\left|f\left(t\right)\right|}^{p}dt\right)}^{\frac{1}{p}}$$

and $L_{[a,b]}^{\infty }$denotes the space of essentially bounded functions on [a, b] with the norm

$${\parallel f\parallel }_{\infty }=\text{ess}\underset{t\in [a,b]}{\sup }\left|f\left(t\right)\right|.$$

In 1934, Grüss in his paper [3] proved that

$$\left|T(f,g)\right|\le \frac{1}{4}(M-m)(N-n),$$

provided that there exist the real numbers m, M, n, N such that

$$m\le f\left(t\right)\le M,n\le g\left(t\right)\le N$$

for a.e. t ∈ [a, b]. The constant 1/4 is the best possible.

In the recent paper [5] M. Niezgoda relaxed the condition of existence of bounding constants in the Grüss theorem to certain class of bounding functions and obtained the following result.

Theorem 1

Let$f,g,\alpha ,\beta ,\gamma ,\delta \in L_{[a,b]}^2$ be functions such that

• α + β and γ + δ are constant functions and

• α(t) ≤ f(t) ≤ β(t) and γ(t) ≤ g(t) ≤ δ(t) for all t ∈ [a, b] or more generally

  $${\int }_a^b(\beta \left(t\right)-f\left(t\right))(f\left(t\right)-\alpha \left(t\right))dt\ge 0\mathit{and}{\int }_a^b(\delta \left(t\right)-g\left(t\right))(g\left(t\right)-\gamma \left(t\right))dt\ge 0.$$

Then we have the inequality

$$\left|T(f,g)\right|\le \frac{1}{4(b-a)}{\parallel \beta -\alpha \parallel }_2{\parallel \delta -\gamma \parallel }_2.$$

M. Niezgoda in [6] gave also a generalization of this result for L^p-spaces.

Theorem 2

Let$f,\alpha ,\beta \in L_{[a,b]}^p$ and$g\in L_{[a,b]}^q$ (1/p + 1/q = 1, 1 ≤ p ≤ ∞) be functions such that

• α + β is a constant function and

• α(t) ≤ f(t) ≤ β(t) for all t ∈ [a, b].

Then we have the inequality

$$\left|T(f,g)\right|\le \frac{1}{2(b-a)}{\parallel \beta -\alpha \parallel }_p{\parallel g-\frac{1}{b-a}{\int }_a^{b}g\left(s\right)ds\parallel }_q.$$

Note that in case p = q = 2 Theorem 1 is the consequence of the Theorem 2.

The aim of this paper is to give weighted generalizations of these results for weighted Čebyšev functional which is given by

$$T_w(f,g)={\int }_a^{b}w\left(x\right)f\left(x\right)g\left(x\right)dx-\left({\int }_a^{b}w\left(x\right)f\left(x\right)dx\right)\left({\int }_a^{b}w\left(x\right)g\left(x\right)dx\right)$$

where w: [a, b] → [0, ∞〉 is some normalized weight function. This is done by using the Sonin’s identity. Generalization of the Grüss inequality with the bounding functions instead of bounding constants in weighted L^p spaces is obtained in Section 2. In the Section 3 this result is utilized to obtain new Ostrowski type inequalities for functions whose derivative belong to L^p spaces. Applications to some weighted numerical rules are given in Section 4.