Abstract
We consider families of general four-point quadrature formulae using a generalization of the Montgomery identity via Taylor’s formula. The results are applied to obtain some sharp inequalities for functions whose derivatives belong to spaces. Generalizations of Simpson’s 3/8 formula and the Lobatto four-point formula with related inequalities are considered as special cases.
Highlights
The most elementary quadrature rules in four nodes are Simpson’s 3/8 rule based on the following four point formula b f t dt a b−a 8 fa 2a b 3 3f a 2b 3 fb −b a f ξ, 1.1 where ξ ∈ a, b, and Lobatto rule based on the following four point formula f t dt −1 1 6 f −1 √ f1 − 2 f6 η, 23625International Journal of Mathematics and Mathematical Sciences where η ∈ −1, 1
Formula 1.1 is valid for any function f with a continuous fourth derivative f 4 on a, b and formula 1.2 is valid for any function f with a continuous sixth derivative f 6 on −1, 1
The obtained formula is used to prove a number of inequalities which give error estimates for the general four-point formula for functions whose derivatives are from Lp-spaces
Summary
The most elementary quadrature rules in four nodes are Simpson’s 3/8 rule based on the following four point formula b f t dt a b−a 8 fa. The obtained formula is used to prove a number of inequalities which give error estimates for the general four-point formula for functions whose derivatives are from Lp-spaces. Identity 2.2 holds true in the case n 1 It can be obtained by taking x ≡ a, x ≡ x, x ≡ a b − x and x ≡ b in 1.5 , multiplying these four formulae by 1/2 − A x , A x , A x and 1/2 − A x , respectively, and adding. For the proof of sharpness, we will find a function f such that b
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