Weighted quadrature rules via Grüss type inequalities for weighted Lp spaces
Introduction
For two functions such that the Čebyšev functional is given by Here, the symbol (1 ≤ p < ∞) denotes the space of p-power integrable functions on the interval [a, b] equipped with the norm and denotes the space of essentially bounded functions on [a, b] with the norm
In 1934, Grüss in his paper [3] proved that
provided that there exist the real numbers m, M, n, N such that 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 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
Then we have the inequality
M. Niezgoda in [6] gave also a generalization of this result for Lp-spaces.
Theorem 2 Let and (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
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 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 Lp 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 Lp spaces. Applications to some weighted numerical rules are given in Section 4.
Section snippets
Grüss type inequalities for weighted Lp spaces
Here and hereafter (1 ≤ p < ∞) denotes the function space of all functions defined on the interval [a, b] bounded with regard to the norm where w: [a, b] → [0, ∞〉 is a normalized weight function, i.e. integrable function satisfying .
In order to obtain the bounds for weighted Čebyšev functional Tw(f, g) for functions which belong to weighted Lp-spaces we need the Sonin’s identity (see [8])
Application to Ostrowski type inequalities
The weighted Montgomery identity for function f (obtained by J. Pečarić in [7]) states that for every x ∈ [a, b]
where the weighted Peano kernel is given by
and w: [a, b] → [0, ∞〉 is a normalized weight function while x ∈ [a, b].
Theorem 5 Let be an open interval, a, b ∈ I, a < b, 1 ≤ p, q ≤ ∞, 1/p + 1/q = 1, w: [a, b] → [0, ∞〉 a normalized weight function,. Then for every and differentiable on I
Application to weighted quadrature rules
Here we apply the results of the previous section to obtain general weighted numerical formula.
Let In: a = t0 < t1 < ⋅⋅⋅ < tn−1 < tn = b be a division of the interval [a, b], ξi ∈ [ti, ti+1] for i = 0, 1, … , n − 1, ξ = (ξ0, ξ1, … , ξn − 1), hk ≔ tk+1 − tk, k = 0, 1, … , n − 1 and ν(h) ≔ max khk. Define the sum where Pw(tk, tk+1; ξk, t) is given by (3.5).
The following approximation theorem holds.
Theorem 7 Let be an
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