NEW ERROR INEQUALITIES FOR THE LAGRANGE INTERPOLATING POLYNOMIAL

A new representation of remainder of Lagrange interpolating 
polynomial is derived. Error inequalities of Ostrowski-Gruss 
type for the Lagrange interpolating polynomial are established. 
Some similar inequalities are also obtained.


Introduction
Many error inequalities in polynomial interpolation can be found in [1,7].These error bounds for interpolating polynomials are usually expressed by means of the norms • p , 1 ≤ p ≤ ∞.Some new error inequalities (for corrected interpolating polynomials) are given in [10,11].The last mentioned inequalities are similar to error inequalities obtained in recent years in numerical integration and they are known in the literature as inequalities of Ostrowski (or Ostrowski-like, Ostrowski-Grüss) type.For example, in [9] we can find inequalities of Ostrowski-Grüss type for the well-known Simpson's quadrature rule, where x i = x 0 + ih, for h > 0, i = 1,2, γ n , Γ n are real numbers such that γ n ≤ f (n) (t) ≤ Γ n , for all t ∈ [x 0 ,x 2 ], and C n are constants, n ∈ {1, 2,3}.
The inequalities of Ostrowski type can be also found in [2,3,4,5,6,12].In some of the mentioned papers, we can find estimations for errors of quadrature formulas which are expressed by means of the differences It is shown that the estimations expressed in such a way can be much better than the estimations expressed by means of the norms As we know there is a close relationship between interpolation polynomials and quadrature rules.Thus, it is a natural try to establish similar error inequalities in polynomial interpolation.
We first establish general error inequalities, expressed by means of f (k) − P m , where P m is any polynomial of degree m and then we obtain inequalities of the above mentioned types.For that purpose, we derive a new representation of remainder of the interpolating polynomial.This is done in Section 2. In Section 3, we obtain the error inequalities of the above-mentioned types.In Section 4, we give some results for derivatives.
Finally, we emphasize that the usual error inequalities in polynomial interpolation (for the Lagrange interpolating polynomial L n (x)) are given by means of the (n + 1)th derivative while in this paper we can find these error inequalities expressed by means of the kth derivative for k = 1,2,...,n.
) and let the assumptions of Lemma 2.1 hold.Then where L n (x) is given by (2.1) and (2.12) Proof.We have (2.13) From (2.13) and (2.6) it follows that since (2.3) holds.
We now suppose that k ≥ 1. Integrating by parts, we obtain (2.16) In a similar way we get (2.17) Continuing in this way, we get (2.18) From (2.14) and (2.18) it follows that

Error inequalities
We now introduce the notations where As we know among all algebraic polynomials of degree ≤ m there exists the only polynomial P * m (t) having the property that where P m ∈ Π m is an arbitrary polynomial of degree ≤ m.We define where C k (•) and E m (•) are defined by (3.2) and (3.6), respectively.
Proof.We set P m (t) = (Γ k+1 + γ k+1 )/2 in (2.12).Then we have (3.12) We also have (3.13) From the above three relations we get The first inequality is proved.We now set P m (t) = γ k+1 in (2.12).Then we have We also have (3.16) Thus, (3.17) The second inequality is proved.In a similar way we prove that the third inequality holds.

.23)
Proof.The proof follows immediately from Theorem 3.3 and Lemma 3.4.

Results for derivatives
Proof.We have (see (2.4)) Thus, We now suppose that for j = 1,2,...,m, m < n − 1 and j + 1 ≤ r ≤ n.We wish to prove that For that purpose, we first calculate (4.8)

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We have by the above assumption.Thus, On the other hand, we have We now rewrite the above relation in the form by the above assumption.We also have where Proof.We define where, obviously, g(t) = f (k+1) (t) − P r (t).We denote see Theorem 2.2.Then we have We introduce the notation

.21)
We now rewrite B(x) in the form

.22)
We have for k ≥ m-see Lemma 4.1.

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We also have On the other hand, we have where E r (•) is defined by (3.6).
be a given subdivision of the interval [a,b] and let f : [a,b] → R be a given function.The Lagrange interpolation polynomial is given by
we choose P r (t) = γ k+1 in Theorem 4.2, then we getf (m) (x) − L (m) n (x) − γ k+1 p (m) ni (x) x − x i k+1 ,(4.34) Remark 3.2.The above estimate has only theoretical importance, since it is difficult to find the polynomial P * .In fact, we can find P * only for some special cases of functions.However, we can use the estimate to obtain some practical estimations-see Theorem 3.3.Theorem 3.3.Let the assumptions of Theorem 2.2 hold.If γ k+1 , Γ k+1 are real numbers such that γ k+1 .9) that is, r ≥ m + 2. Thus (4.7) holds.This completes the proof.