## Abstract

A general interpolating formula is established. From this formula all Newton–Cotes quadrature rules of the closed type can be derived. Some corrected interpolating polynomials are also derived and used for obtaining corresponding quadrature rules. A new effective representation of the Peano kernel is derived. Estimation of errors for these quadrature rules is established.

## 1. Introduction

In recent years, a number of authors have considered error analyses of some known and new quadrature rules (Fedotov & Dragomir 1999; Pearce & Pečarić 2000; Pearce *et al*. 2000; Cerone 2001; Cheng 2001; Pečarić & Varošanec 2001; Dragomir 2003; Engelbrecht *et al*. 2003; Matić 2003). Such rules can be obtained by integrating interpolation polynomials. In the work of Pečarić & Ujević (submitted), we can find a new general and a few particular corrected interpolating polynomials, including the general formula(1.1)where(1.2)and is a polynomial, such that , is the Lagrange interpolating polynomial, are basic Lagrange polynomials, are Appell polynomials and is a given division of the interval . It is shown in the work of Pečarić & Ujević (submitted) that the Lagrange interpolating polynomial is merely a special case of the above formula. The authors also give the particular formula(1.3)(1.4)where , are Bernoulli polynomials and are Bernoulli numbers, while is the divided difference of order *m* for the function . A similar formula for Euler polynomials is obtained in the work of Ujević (2004). These interpolating polynomials can be used for obtaining quadrature rules in the usual way.

In this paper, we give different (corrected) interpolating polynomials and these are used to obtain some known and some new generalized quadrature rules. As a consequence of a general interpolating formula, we can derive all Newton–Cotes formulae of the closed type. We also give a very effective (piecewise integral) representation of the Peano kernel for all obtained quadrature rules. Finally, we illustrate applications of the obtained results, deriving some error inequalities.

## 2. Identities and Peano kernel representation

Let be a given division of the interval and let be a given function. The Lagrange interpolation polynomial is given by(2.1)where(2.2)for . and , , are as above with the Cauchy relations (Mhaskar & Pai 2000)(2.3)and(2.4)

Let be a given uniform division of the interval , i.e. , , . The Lagrange interpolating polynomial is then given bywhere , .

The divided difference of the first order of the function *f* is given byand the divided difference of order *n* defined by the recurrence formula

*The* *nth-order divided difference satisfies the relation*

The interpolating polynomial can be written in the Newton form aswhere(2.5)for (Agarwal & Wong 1993).

Also defined is a harmonic (or Appell) sequence of polynomials , i.e. polynomials which satisfy , and which can be expressed as(2.6)We use the notation

*Let* *be a harmonic (or Appell) sequence of polynomials*. *Let* *belong to* *, such that* *and let n be an arbitrary (but fixed) positive integer. It then follows that*(2.7)*where*(2.8)*for any* . *If k*=0,

We prove that (2.7) holds for by induction. For *k*=0,since (2.3) holds. Withsuch that(2.9)we need to prove that(2.10)Nowandsuch thatHence, (2.10) holds. ▪

The identities (1.1) and (2.7) are similar. Note, however, that (1.1) holds for , while (2.7) holds for any . Note also that (2.7) can be interpreted as an interpolation formula.

Consider now quadrature formulae obtained using the above interpolation. For this purpose, we integrate from to , giving(2.11)Sinceis a Newton–Cotes quadrature formula, if we suppose that the partition *D* is uniform, we then obtain a corrected Newton–Cotes formula.

We now introduce some additional notation in order to simplify subsequent developments:whereandsuch thatFurther,andAlso, let

*Under the assumptions of* *theorem 2.2**,*(2.12)*and*(2.13)*where* , *if p*>1 *or q*>1 *and if p*=1*, then* *and if* *, then* .

We haveWe also havesuch thatIn other words,Equation (2.13) is a consequence of Hölder's inequality. ▪

Note that the function is a new representation of the Peano kernel. This is the main result of this paper. It can be used for deriving Peano kernels for different quadrature rules using Maple (or Matlab or Mathematica). It can also be used for deriving error bounds for these rules (Roumeliotis 2003). We illustrate this ‘manually’ in §3.

In principle, we can substitute any sequence of polynomials of the form (2.6) in (2.11), but, of course, we seek sequences that provide effective results. Here, we choose the following polynomials which were introduced in the work of Matić *et al*. (1999):(2.14)(2.15)(2.16)(2.17)where and are Bernoulli and Euler polynomials, respectively. Properties of Bernoulli and Euler polynomials that we use here can be found in Abramowitz & Stegun (1965).

*Under the assumptions of* *theorem 2.2**,**where**for* .

Substitute (2.14) in theorem 2.2 and note thatfor , since (2.4) holds. ▪

Now consider the quadrature formula(2.18)i.e. if we suppose that the underlying partition *D* is uniform, then we consider all Newton–Cotes formulae of the closed type.

*Under the assumptions of* *theorem 2.2**,*(2.19)*where*(2.20)*and*

Substitute (2.15) in theorem 2.2 and note that ▪

We also consider the generalized quadrature formula(2.21)

*Under the assumptions of* *theorem 2.2**,*(2.22)*for* *,* *where*(2.23)*and*

Substitute (2.16) in theorem 2.2 and note thatfor andby lemma 2.1. ▪

In this case, we consider the quadrature formula(2.24)

If we take (2.24) for *n*=1, *k*=1, then we havewith . We now calculateandHence,(2.25)which is the well-known corrected trapezoidal rule.

*Under the assumptions of* *theorem 2.2**,**for* *, where**and*

The proof is similar to that of corollary 2.8. Here, we substitute the polynomials (2.17). ▪

Finally, consider(2.26)

If we write this for *n*=1, *k*=1, then we again obtain the corrected trapezoidal rule. However, further generalization of this is different from the generalization given in (2.24).

## 3. Error inequalities

Here, we illustrate applicability of the results obtained in §2, confining our attention to a few specially chosen cases.

Theorem 2.4 provides opportunity to find error bounds for all Newton–Cotes formulae of the closed type. For example, we can find error inequalities for the trapezoidal quadrature rule. Here, we also demonstrate the applicability of theorem 2.4 to a generalized trapezoidal rule, the corrected trapezoidal rule and Simpson's rule.

### (a) Trapezoidal rule

If we choose *n*=1 and *k*=1 in (2.18), then we get the trapezoidal quadrature rulewherewith , . We can now use theorem 2.4 to obtain error bounds for this rule. We haveandwhere .

For the sake of simplicity, further considerations (calculations) are made in the interval . We havesuch thatThus,where is the beta function. Sincewhere is the gamma function , we can writeIn particular, if *q* is a positive integer, thensince in this case.

In the interval , we have the estimate(3.1)For *q*=1, we get the well-known classical estimate(3.2)

From the work of Dragomir *et al*. (2000),which can be written as

### (b) A generalized trapezoidal rule

We now consider (2.21) of corollary 2.7 and choose *n*=1, . In fact, we consider a generalization of the trapezoidal quadrature rule.

First, we write the generalized quadrature rule for the above choice of *n* and *k* , givingandsuch that we have to calculateandA simple verification shows thatandWe have thenThe remainder term (error) is now estimated using theorem 2.4. We havegivingandThus,If , , thenand if , , thensuch thatandrespectively.

### (c) Corrected trapezoidal rule

We now apply theorem 2.4 to the corrected trapezoidal quadrature rule (2.25). Again, for the sake of simplicity, all calculations are first performed for the interval . We haveandsince , and . Hence,andsuch thatWe have thenSince this constant is equal for all intervals, finally we get(3.3)

This result is also obtained in the work of Dedić *et al*. (2001) using a different approach. In the work of Dragomir *et al*. (2000), we also find the inequalitywhere

### (d) Simpson's rule

Finally, theorem 2.4 is applied to the well-known Simpson's rule. If we substitute *n*=2 in (2.18), thenwhere , , . Here, we choose *k*=0. Again, for , we haveorwhere are as defined previously. A simple calculation givessuch thatSince this constant is equal for all intervals, we havewhere .

This result is also obtained in the work of Pečarić & Varošanec (2001) using a different approach.

## Footnotes

- Received September 20, 2005.
- Accepted March 14, 2006.

- © 2006 The Royal Society