# A general interpolating formula and error bounds.

1. Introduction

Polynomial interpolation is the interpolation of a given data set by a polynomial. As we know there is a unique interpolating polynomial which can be written in different forms. Here we use the Lagrange form.

The main result of this paper is a general interpolating formula. We show that the Lagrange interpolating polynomial is only a special case of this general formula. Using this formula we can obtain many particular interpolating formulas. In this paper we use the formula to obtain few corrected interpolating polynomials, as an illustration. Some corrected interpolating polynomials are given in (11) and (12). In (12) we can find the following corrected interpolating formula

f(x) = [L.sub.n](x) + [[omega].sub.n](x)[[[k + 1]/2].summation over (j = 1)][2/[[2.sup.[2j - 1]](2j - 1)!]][g.sub.[2j - 1]][[x.sub.0];[x.sub.1]; ...;[x.sub.n]] + [[tilde.R].sub.k](x),

where [L.sub.n](x) is the Lagrange interpolating polynomial, [[omega].sub.n] (x) = (x-[x.sub.0]) ... (x-[x.sub.n]), [g.sub.m](t) = ([x-t).sup.[m-1]] f.sup.(m)]([x+t]/2), [g.sub.m] [[x.sub.0]; [x.sub.1]; ...; [x.sub.n]] denote the divide difference of order m and [R.sub.k](x) is a remainder term. We also suppose that we have a given partition of the interval [a, b] ([DELTA] = {a = [x.sub.0] < ... < [x.sub.n] = b}) and that f [member of] [C.sup.k+1] (a, b).

In (11) we can find the following formula

f(x) = [L.sub.n](x) + [[omega].sub.n](x)[[[k - 1]/2].summation over (j = 0)][[[E.sub.[2j + 1]](0)]/[(2j + 1)!]][g.sub.[2j - 1]][[x.sub.0];[x.sub.1]; ...;[x.sub.n]] + [R.sub.k](x),

where [E.sub.j](x) are Euler polynomials, [g.sub.m](t) = [(x-t).sup.m-1] f.sup.(m)](t) and [R.sub.k](x) is a remainder term. (Note that [g.sub.m](t) = [g.sub.m](t, x). We use this simplification in the rest of this paper.) We show that the above formula is a special case of the mentioned general interpolating formula. In (11) many error inequalities for the above formula are derived. Here we improve some of these inequalities.

In section 2 we give a general interpolating formula. In section 3 we give few particular corrected interpolating polynomials. We specially mention Corollary 5 in which we show that the Lagrange interpolating polynomial is only a special case of the general formula. In section 4 we give some general error inequalities for the general interpolating formula. In section 5 we give various error bounds for these corrected interpolating polynomials. Similar error inequalities are obtained in numerical integration. For example see (3-6), (9) and (10). In some of these paper it is shown that the obtained error inequalities are superior to bounds obtained in more standard ways. Thus we use these kind of inequalities in this paper to obtain error bounds for interpolating polynomials.

Finally, we emphasize that the usual error inequalities in polynomial interpolation (for the Lagrange interpolating polynomial [L.sub.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.

2. A General Corrected Interpolating Polynomial

Let [DELTA] = {a = [x.sub 0] < [x.sub.1] < ... < [x.sub.n] = b} be a given subdivision of the interval [a, b] and let f: [a, b] [right arrow] R be a given function. The Lagrange interpolation polynomial is given by

[L.sub.n](x) = [n.summation over (i = 0)][p.sub.ni](x)f([x.sub.i]), (1)

where

[p.sub.ni](x) = [[(x - [x.sub.0]) ... (x - [x.sub.[i - 1]])(x - [x.sub.[i + 1]]) ... (x - [x.sub.n])]/[([x.sub.i] - [x.sub.0]) ... ([x.sub.i] - [x.sub.[i - 1]])([x.sub.i] - [x.sub.[i + 1]]) ... ([x.sub.i] - [x.sub.n])]] (2)

for i = 0, 1, ..., n. Here we always use the notation [L.sub.n](x) for the Lagrange interpolating polynomial and pni(x), i = 0, 1, 2, ..., n, denote the basic Lagrange interpolating polynomials. We have the Cauchy relations ([7, pp. 160-161]),

[n.summation over (i = 0)][p.sub.ni](x) = 1 (3)

and

[n.summation over (i = 0)][p.sub.ni](x)[[(x - [x.sub.i])].sup.j] = 0,j = 1,2, ...,n. (4)

Let [DELTA] = {[x.sub.0] = a < [x.sub.1] < ... < [x.sub.n] = b} be a given uniform subdivision of the interval [a, b], i.e. [x.sub.i] = [x.sub 0] + ih, h = (b-a)/n, i = 0, 1, 2, ..., n. Then the Lagrange interpolating polynomial is given by

[L.sub.n](x) = [L.sub.n]([x.sub.0] + th) = [[( - 1)].sup.n][[t(t - 1) ... (t - n)]/[n!]][n.summation over (i = 0)][[( - 1)].sup.i](ni)[[f([x.sub.i)]]/[t - i]],

where t [??] {0, 1, 2, ..., n}, 0 < t < n.

As we know the divided difference of the first order of the function f is given by

f[[x.sub.0];[x.sub.1]] = [[f([x.sub.1]) - f([x.sub.0])]/[x1 - x0]].

The divided difference of order n is defined via the divided differences of order n-1 by the recurrence formula

f[[x.sub.0];[x.sub.1]; ...;[x.sub.n]] = [[f[[x.sub.1];[x.sub.2]; ...;[x.sub.n]] - f[[x.sub.0];[x.sub.1]; ...;[x.sub.[n - 1]]]]/[[x.sub.n] - [x.sub.0]]].

The following lemma is valid ([2, p. 68]).

Lemma 1. The nth-order divided difference satisfies the relation

f[[x.sub.0];[x.sub.1]; ...;[x.sub.n]] = [n.summation over (i = 0)][[f([x.sub.i])]/[([x.sub.i] - [x.sub.0]) ...([x.sub.i] - [x.sub.[i - 1]])([x.sub.i] - [x.sub.[i + 1]]) ...([x.sub.i] - [x.sub.n])]].

The interpolating polynomial can be written in the Newton form as

[L.sub.n](x) = f([x.sub.0]) + [[n = 1].summation over (i = 0)](x - [x.sub.0]) ...(x - [x.sub.i])f[[x.sub.0]; ...;[x.sub.[i + 1]]]

= f([x.sub.0]) + [[n - 1].summation over (i = 0)][[omega].sub.i](x)f[[x.sub.0]; ...;[x.sub.[i + 1]]],

where

[[omega].sub.i](x) = (x - [x.sub.0])(x - [x.sub.1]) ... (x - [x.sub.i]), (5)

for i = 0, 1, 2, ..., n.

Lemma 2. Let [P.sub.m](t) be any polynomial of degree [less than or equal to] m and let [DELTA] be a given partition of the interval [a, b]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

for 0 [less than or equal to] m [less than or equal to] n-1, where pni(x) are basic Lagrange polynomials and x [member of] [a, b].

Proof. Let x be a real number. Then we have

[P.sub.m](t) = [m.summation over (j = 0)][c.sub.j][[(x - t)].sup.j],

for some coefficients [c.sub.j] = [c.sub.j](x), j = 0, 1, ...,m. (This is a consequence of the Taylor formula.) Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It follows that

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Finally, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for 0 [less than or equal to] m [less than or equal to] n-1, since (4) holds.

In what follows (to simplify notations and records) we denote

[P.sub.j](t, [x.sub.i]) = [P.sup.j](t)

such that

[P.sub.j.sup.(m)](t) = [[[[partial derivative].sup.m][P.sub.j](t,[x.sub.i])]/[[partial derivative][t.sup.m]]],

where [x.sub.i] [member of] [a, b].

Theorem 3. Under the assumptions of Lemma 2 suppose that f [member of] [C.sup.k+1](a, b). Let {[P.sub.k](t)} be a harmonic (or Appell) sequence of polynomials, i.e. [P'.sub.k](t) = [P.sub.k-1](t), [P.sub.0](t) = 1. Then

f(x) = [L.sub.n](x) - [k.summation over (m = 1)][[( - 1)].sup.m][n.summation over (i = 0)][p.sub.ni](x)[[P.sub.m](x)[f.sup.(m)](x) - [P.sub.m]([x.sub.i])[f.sup.(m)]([x.sub.i])] + [R.sub.k,j](x), (7)

where

[R.sub.k,j](x) = [[( - 1)].sup.k][n.summation over (i = 0)][p.sub.ni](x)[[integral].sub.[x.sub.i].sup.x][[f.sup.[(k + 1)]](t) - [Q.sub.j](t)][P.sub.k](t)dt, (8)

for any polynomial [Q.sub.j](t) such that 0 [less than or equal to] j + k [less than or equal to] n-1.

Proof. Integrating by parts, we obtain

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In a similar way we have

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Continuing in this way we get

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Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we define [S.sub.m](t) = [Q.sub.j](t)[P.sub.k](t), m = j + k, then we also have

[[( - 1)].sup.k][n.summation over (i = 0)][p.sub.ni](x)[[integral].sub.[x.sub.i].sup.x][Q.sub.j](t)[P.sub.k](t)dt = [[( - 1)].sup.k][n.summation over (i = 0)][p.sub.ni](x)[[integral].sub.[x.sub.i].sup.x][S.sub.m](t)dt = 0,

for 0 [less than or equal to] m [less than or equal to] n-1, since (6) holds.

From the above two relations we easily conclude that (7) is valid.

Remark 4. Note that we can choose k = 0 in the above theorem. In this case we have no perturbation of the Lagrange interpolating polynomial. (Of course, the corresponding sum in (7) is empty, if k = 0.) If [Q.sub.j](t) = 0 then we can set k [less than or equal to] n.

3. Particular Corrected Interpolating Polynomials

If we substitute the polynomials

[P.sub.k](t) = [[[(t - [x.sub.i])].sup.k]/[k!]]

in Theorem 3 then we get the following result. (Recall that we have adopted the notation [P.sub.k](t) = [P.sub.k](t, [x.sub.i])-mostly to simplify records.)

Corollary 5. Under the assumptions of Theorem 3 we have

f(x) = [L.sub.n](x) + [R.sub.[k,j]](x), (9)

where [R.sub.[k,j](x) is given by (8).

Hence, we got the Lagrange interpolating polynomial (without perturbation). See also Remark 4.

If we substitute the polynomials

[P.sub.k](t) = [[[(x - [x.sub.i])].sup.k]/[k!]][E.sub.k]([t - [x.sub.i]]/[x - [x.sub.i]]), (10)

([E.sub.k](t) are Euler polynomials) in Theorem 3 then we get the following corrected interpolating polynomial.

Corollary 6. Under the assumptions of Theorem 3 we have

f(x) = [L.sub.n](x) + [[omega].sub.n](x)[k.summation over (m = 1)][[[[( - 1)].sup.m][E.sub.m](0)]/[m!]][g.sub.m][[x.sub.0];[x.sub.1]; ...[x.sub.n]] + [R.sub.k,j](x), (11)

where[[omega].sub.n](x) = (x - [x.sub.0]) ... (x - [x.sub.n]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

and

[g.sub.m](t) = [(x-t).sup.[m-1]][f.sup.(m)](t), m = 1, 2, ..., k.

This result is proved in [11]. (Recall that we have adopted the simplification [g.sub.m](t, x) = [g.sub.m](t).)

It is obvious that we can substitute infinitely many harmonic sequences of polynomials in Theorem 3 to obtain various corrected interpolating formulas.

Here we consider the polynomials

[P.sub.k](t) = [[[(x - [x.sub.i])].sup.k]/[k!]][B.sub.k]([t - [x.sub.i]]/[x - [x.sub.i]]), (13)

where [B.sub.k](t) are Bernoulli polynomials. For these polynomials we can get applicable results.

We now recall some properties of Bernoulli polynomials. The Bernoulli polynomials are defined by the relation

[[t[e.sup.xt]]/[[e.sup.t] - 1]] = [[[infinity]].summation over (k = 0)][B.sub.k](x)[[t.sup.k]/[k!]],|t|<2[pi],

such that

[B.sub.0](x) = 1,[B.sub.1](x) = x - [1/2],[B.sub.2](x) = [x.sup.2] - x + [1/6], ... (14)

We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

|[B.sub.2n](t)|[less than or equal to]|[B.sub.2n]|,n = 1,2, ... (18)

0<[[( - 1)].sup.n][B.sub.[2n + 1]](t)<[[2(2n + 1)!]/[(2[pi]).sup.[2n + 1]]](1/[1 - [2.sup.[ - 2n]]]),0<x<[1/2],n = 1,2, ... (19)

The numbers [B.sub.k] = [B.sub.k](0) are Bernoulli numbers. We have [B.sub.0] = 1, [B.sub.1] = -1/2, [B.sub.2] = 1/6, ..., [B.sub.2j+1] = 0 for j = 1, 2, ... and [B.sub.k](0) = [(-1).sup.k][B.sub.k](1). (20)

Further properties of Bernoulli polynomials can be found in (1).

If we substitute the polynomials (13) in Theorem 3 then we get the following result.

Corollary 7. Under the assumptions of Theorem 3 we have

f(x) = [L.sub.n](x) + [[omega].sub.n](x)[k.summation over (m - 1)][[[[( - 1)].sup.m][B.sub.m]]/[m!]][g.sub.m][[x.sub.0];[x.sub.1]; ...;[x.sub.n]] + [R.sub.k,j](x), (21)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

and

[g.sub.m](t) = [(x-t).sup.[m-1]] [f.sup.(m)](t), m = 1, 2, ..., k.

Proof. If [P.sub.m](t) are defined by (13) then we have

[P.sub.m](x)[f.sup.(m)](x) - [P.sub.m]([x.sub.i])[f.sup.(m)]([x.sub.i]) = [[[(x - [x.sub.i])].sup.m]/[m!]][[B.sub.m](1)[f.sup.(m)](x) - [B.sub.m](0)[f.sup.(m)]([x.sub.i])]

such that

[k.summation over (m = 1)][[( - 1)].sup.m][n.summation over (i = 0)][p.sub.ni](x)[[P.sub.m](x)[f.sup.(m)](x) - [P.sub.m]([x.sub.i])[f.sup.(m)]([x.sub.i]) = [k.summation over (m = 1)][[[( - 1)].sup.m]/[m!]][n.summation over (i = 0)][p.sub.ni](x)[[(x - [x.sub.i])].sup.m][[B.sub.m](1)[f.sup.(m)](x) - [B.sub.m](0)[f.sup.(m)]([x.sub.i])] = [k.summation over (m = 1)][[[( - 1)].sup.m]/[m!]][B.sub.m](0)[n.summation over (i = 0)][p.sub.ni](x)[[(x - [x.sub.i])].sup.m][f.sup.(m)]([x.sub.i]),

since

[n.summation over (i = 0)][p.sub.ni](x)[[(x - [x.sub.i])].sup.m][B.sub.m](1)[f.sup.(m)](x) = 0,

for 1 [less than or equal to] m [less than or equal to] n. We have

[n.summation over (i = 0)][p.sub.ni](x)[[(x - [x.sub.i])].sup.m][f.sup.(m)]([x.sub.i]) = [[omega].sub.n](x)[n.summation over (i = 0)][[[[(x - [x.sub.i])].sup.[m - 1]][f.sup.(m)]([x.sub.i])]/[([x.sub.i] - [x.sub.0]) ... ([x.sub.i] - [x.sub.[i - 1]])([x.sub.i] - [x.sub.[i + 1]]) ... ([x.sub.i] - [x.sub.n])]]

and

[g.sub.m][[x.sub.0];[x.sub.1]; ... ;[x.sub.n]] = [n.summation over (i = 0)][[[[(x - [x.sub.i])].sup.[m - 1]][f.sup.(m)]([x.sub.i])]/[([x.sub.i] - [x.sub.0]) ... ([x.sub.i] - [x.sub.[i - 1]])([x.sub.i] - [x.sub.[i + 1]]) ... ([x.sub.i] - [x.sub.n])]],

by Lemma 1, such that

[k.summation over (m = 1)][[( - 1)].sup.m][n.summation over (i = 0)][p.sub.ni](x)[[P.sub.m](x)[f.sup.(m)](x) - [P.sub.m]([x.sub.i])[f.sup.(m)]([x.sub.i])] = [[omega].sub.n](x)[k.summation over (m = 1)][[[( - 1)].sup.m]/[m!]][B.sub.m](0)[g.sub.m][[x.sub.0];[x.sub.1]; ... ;[x.sub.n]].

From (7), (8) and the above relation we see that (21) holds.

Remark 8. Since [B.sub.2m+1] = 0, m = 1, 2, ..., and [B.sub.1] = -1/2 we can write

f(x) = [L.sub.n](x) + [1/2][[omega].sub.n](x)[g.sub.1][[x.sub.0];[x.sub.1]; ...;[x.sub.n]] + [[omega].sub.n](x)[[k/2].summation over (m = 1)][[B.sub.2m]/[(2m)!]][g.sub.2m][[x.sub.0];[x.sub.1]; ...;[x.sub.n]] + [R.sub.k,j](x).

We also consider the polynomials

[P.sub.k](t) = [[[(t - x)].sup.k]/[k!]]. (23)

Corollary 9. Under the assumptions of Theorem 3 we have

f(x) = [L.sub.n](x) + [[omega].sub.n](x)[k.summation over (m = 1)][1/[m!]][g.sub.m][[x.sub.0];[x.sub.1]; ...;[x.sub.n]] + [R.sub.k,j](x), (24)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

and

[g.sub.m](t) = [(x-t).sup.[m-1]] [f.sup.(m)](t), m = 1, 2, ..., k.

Proof. If we substitute the polynomials (23) in Theorem 3 then we get

f(x) = [L.sub.n](x) + [k.summation over (m = 1)][[( - 1)].sup.m][n.summation over (i = 0)][p.sub.ni](x)[P.sub.m]([x.sub.i])[f.sup.(m)]([x.sub.i]) + [R.sub.k,j](x) = [L.sub.n](x) + [k.summation over (m = 1)][[[( - 1)].sup.m]/[m!]][n.summation over (i = 0)][p.sub.ni](x)[[([x.sub.i] - x)].sup.m][f.sup.(m)]([x.sub.i]) + [R.sub.k,j](x),

since [P.sub.m](x) = 0 and

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The above formula can be written in the form

f(x) = [L.sub.n](x) + [k.summation over (m = 1)][1/[m!]][n.summation over (i = 0)][p.sub.ni][[(x - [x.sub.i])].sup.m][f.sup.(m)]([x.sub.i]) + [R.sub.k,j](x) = [L.sub.n](x) + [[omega].sub.n](x)[k.summation over (m = 1)][1/[m!]][g.sub.m][[x.sub.0];[x.sub.1]; ...;[x.sub.n]],

where

[g.sub.m](t) = [[(x - t)].sup.[m - 1]][f.sup.(m)](t),m = 1,2, ...,k.

4. General Error Inequalities

Let g [member of] C(a, b). As we know among all algebraic polynomials of degree < j there exists the only polynomial [Q*.sub.j](t) having the property that

[parallel]g - [Q.sub.j.sup.*][[parallel].sub.[infinity]][less than or equal to] [parallel]g - [Q.sub.j][[parallel].sub.[infinity]],

where [Q.sub.j] [member of] [[PI].sub.j] is an arbitrary polynomial of degree < j. We define

[G.sub.j](g) = [parallel]g - [Q.sub.j.sup.*][parallel] = [[inf][Q.sub.j][member of][product.sub.j]][parallel]g - [Q.sub.j][[parallel].sub.[infinity]]. (26)

Here we use the above notation [G.sub.j] without referring to its definition.

We also introduce the notations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

Specially, if [[x.subi], x] = [0, 1] then we use the notations [parallel]g[[parallel].sub.[[infinity],1]] and [parallel]g[[parallel].sub.[1,1]]

Theorem 10. Under the assumptions of Theorem 3 we have

|f(x) = [L.sub.n](x) + [k.summation over (m = 1)][[( - 1)].sup.m][n.summation over (i = 0)][p.sub.ni](x)[[P.sub.m](x)[f.sup.(m)](x) - [P.sub.m]([x.sub.i])[f.sup.(m)](x)]| (29)

[less than or equal to] [G.sub.j]([f.sup.[(k + 1)]])[n.summation over (i = 0)]|[p.sub.ni](x)[parallel]|[P.sub.k][[parallel].sub.[1,i]].

Proof. Let [Q.sup.j](t) = [Q.sub.j.sup.*](t), where [Q.sub.j.sup.*](t) is defined by (26) for the function g(t) = [f.sup.(k+1)](t). Then we have

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From (7), (8), (28) and the above relation we see that (29) holds.

Theorem 11. Let the assumptions of Theorem 3 hold. If [gamma]k+1, [[GAMMA].sub.k+1] are real numbers such that [gamma]k+1 [less than or equal to] [f.sup.(k+1](t) [less than or equal to] [[GAMMA].sub.k+1], t [member of] [a, b], k = 0, 1, ..., n-1, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

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where [S.sub.ki] =[[f.sup.(k)](x)-f.sup.(k)]([x.sub.i]]/(x-xi), i = 0, 1, ..., n.

Proof. We set [Q.sub.j](t) = ([[GAMMA].sub.k+1] + [gamma]k+1) /2 in (8). Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since

[parallel][f.sup.[(k + 1)]] - [[[[GAMMA].sub.[k + 1]] + [[gamma].sub.[k + 1]]]/2][[parallel].sub.[infinity]][less than or equal to] [[[[GAMMA].sub.[k + 1]] - [[gamma].sub.[k + 1]]]/2]

we get

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The first inequality is proved.

We now have

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since

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The second inequality is proved. The third inequality can be proved in a similar way.

5. Particular Error Inequalities

In this section we give various error bounds for the particular corrected interpolating polynomials.

For that purpose, we introduce the notations

[C.sub.k](x) = [n.summation over (i = 0)][[|x - [x.sub.i][|.sup.k]]/[|[x.sub.i] - [x.sub.0]|...|[x.sub.i] - [x.sub.[i - 1]][parallel][x.sub.i] - [x.sub.[i + 1]]| ... |[x.sub.i] - [x.sub.n]|]],(30)

[F.sub.k](x) = [n.summation over (i = 0)][[([S.sub.ki] - [[lambda].sub.[k + 1]])|x - [x.sub.i][|.sup.k]]/[|[x.sub.i] - [x.sub.0]| ... |[x.sub.i] - [x.sub.[i - 1]][parallel][x.sub.i] - [x.sub.[i + 1]]| ... |[x.sub.i] - [x.sub.n]|]],(31)

[D.sub.k](x) = [n.summation over (i = 0)][[([[GAMMA].sub.[k + 1]] - [S.sub.ki])|x - [x.sub.i][|.sup.k]]/[|[x.sub.i] - [x.sub.0]| ... |[x.sub.i] - [x.sub.[i - 1]][parallel][x.sub.i] - [x.sub.[i + 1]]| ... |[x.sub.i] - [x.sub.n]|]], (32)

where [S.sub.ki] = [f.sup.(k)](x)-[f.sup.(k)]([x.sub.i])/(x-[x.sub.i]), i = 0, 1, ..., n and [gamma]k+1, [[GAMMA]k+1] are real numbers such that [gamma]k+1 < [f.sup.(k+1](t) < [GAMMA].sub.k+1], t [member of] [a, b], k = 0, 1, ..., n-1.

We also introduce notations for the remainders of the corrected interpolating polynomials. These interpolating polynomials are obtained in Section 3. We define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

In this section we use the above notations without referring to their definitions.

We now consider the interpolation formula given in Corollary 7.

Theorem 12. Under the assumptions of Corollary 7 we have

|[R.sub.[k,j].sup.B](x)|[less than or equal to] [[[parallel][B.sub.k][[parallel].sub.1,1]]/[k!]][G.sub.j]([f.sup.[(k + 1)]])[C.sub.k](x)|[[omega].sub.n](x)|.

Proof. We substitute the polynomials (13) in Theorem 10. If we use the fact that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

then we easily find that the above inequality holds.

Corollary 13. Under the assumptions of Theorem 12 let k be odd, k = 2r - 1, r [greater than or equal to] 1. Then we have

|[R.sub.[k,j].sup.B](x)|[less than or equal to] [[2|[B.sub.2r]|]/[(2r - 1)!]][[1 - [2.sup.[ - 2r]]]/r][G.sub.j]([f.sup.(2r)])[C.sub.[2r - 1]](x)|[[omega].sub.n](x)|,

If k is even then we have

|[R.sub.[k.j].sup.B](x)|[less than or equal to] [[2[[LAMBDA].sub.k]]/[(k + 1)!]][G.sub.j]([f.sup.[(k + 1)]])[C.sub.k](x)|[[omega].sub.n](x)|, (37)

where

[[LAMBDA].sub.k] = 2|[B.sub.[k + 1]]([t.sub.0])| (38)

and [t.sub.0] [member of] (0, 1/2) is a unique zero point (in the interval (0,1/2)) of the Bernoulli polynomial [B.sub.k](*), k = 2r.

Proof. To prove the first inequality it is sufficient to note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (39)

if k [greater than or equal to] 2 is odd, k = 2r-1. (The last integral is calculated in (6).)

To prove the second inequality we calculate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

Remark 14. The above estimates have only theoretical importance, since it is difficult to find the polynomial Q* in practice. In fact, we can find Q* only for some special cases of functions. However, we can use the estimates to obtain some practical estimations.

We also have the estimation

|[R.sub.[k,j].sup.B](x)|[less than or equal to][[square root of [|[B.sub.2k]|]]/[square root of [(2k)!]]][G.sub.j]([f.sup.[(k + 1)]])[C.sub.k](x)|[[omega].sub.n](x)|.

This estimation follows from Theorem 12 and the fact that

[([[integral].sub.0.sup.1]|[B.sub.k](t)|dt).sup.2][less than or equal to][[integral].sub.0.sup.1][([B.sub.k](t)).sup.2]dt = [[(k!).sup.2]/[(2k)!]]|[B.sub.2k]|. (41)

We also emphasize that this inequality would be used only if k is even and we cannot find (or we simply don't know) the zero point [t.sub.0].

Finally, we specially note that if k is even then the corresponding Bernoulli polynomial [B.sub.k](t) has only one zero point [t.sub.0] in (0, 1/2).For example, for k = 2 this zero point is [t.sub.0] = [1/2 - 1/6][square root of.3] for k = 4 the zero point is [t.sub.0] = [1/2 - 1/30] [square root of (225 - 30][square root of.30]and generally, for large k = 2r, it is (very) close to 1/4.

Theorem 15. Let the assumptions of Corollary 7 hold. If [[gamma].sub.k+1], [[GAMMA].sup.k+1] are real numbers such that [[gamma].sub.k+1] [less than or equal to] [f.sup.(k+1](t) [less than or equal to] [GAMMA].sub.k+1], t [member of] [a, b], k = 0, 1, ..., n-1, then

|[R.sub.[k,j].sup.B](x)|[less than or equal to][[[[GAMMA].sub.[k + 1]] - [[gamma].sub.[k + 1]]]/[2k!]][C.sub.k](x)|[[omega].sub.n](x)[parallel]|[B.sub.k][[parallel].sub.1,1],

where [[omega].sub.n](x) is defined by (5). We also have

|[R.sub.[k,j].sup.B](x)|[less than or equal to][[|[[omega].sub.n](x)]/[k!]][F.sub.k](x)[parallel][B.sub.k][[parallel].sub.[infinity],1]

and

|[R.sub.[k,j].sup.B](x)|[less than or equal to][[|[[omega].sub.n](x)|]/[k!]][D.sub.k](x)[parallel][B.sub.k][[parallel].sub.[infinity],1].

Proof. We substitute the polynomials (13) in Theorem 11 and use the relation (36).

Corollary 16. Let the assumptions of Theorem 15 hold. If k is odd, k = 2r-1, r [greater than or equal to] 2, then

|[R.sub.[k,j].sup.B](x)|[less than or equal to][[[[GAMMA].sub.[k + 1]] - [[gamma].sub.[k + 1]]]/[(2r - 1)]][[1 - [2.sup.[ - 2r]]]/r]|[B.sub.2r]|[C.sub.k](x)|[[omega].sub.n](x)|.

If k is even then we have

|[R.sub.[k,j].sup.B](x)|[less than or equal to][[[[GAMMA].sub.[k + 1]] - [[gamma].sub.[k + 1]]]/[(k + 1)!]][[LAMBDA].sub.k][C.sub.k](x)|[[omega].sub.n](x)|

where [[LAMBDA].sub.k] is defined by (38).

Proof. The proof follows from the above theorem and the relations (39) and (40).

Remark 17. We additionally have the estimate

|[R.sub.[k,j].sup.B](x)|[less than or equal to][[[[GAMMA].sub.[k + 1]] - [[gamma].sub.[k + 1]]]/[2[square root of [(2k)]]!]][square root of [|[B.sub.2k]|]][C.sub.k](x)|[[omega].sub.n](x)|. (42)

which can be proved using the relation (41). See also Remark 14.

Corollary 18. Let the assumptions of Theorem 15 hold. Then

|[R.sub.[k,j].sup.B](x)|[less than or equal to][2/[(2[pi]).sup.k]][1/[1 - [2.sup.[1 - k]]]]|[[omega].sub.n](x)|[F.sub.k](x),

if k = 4r + 3, r = 0, 1, 2, ...,

|[R.sub.[k,j].sup.B](x)|[less than or equal to][[|[B.sub.k]|]/[k!]]|[[omega].sub.n](x)|[F.sub.k](x),

if k = 2r, r = 1, 2, ...,

|[R.sub.[k,j].sup.B](x)|[less than or equal to][2/[(2[pi]).sup.k]][1/[1 - [2.sup.[1 - k]]]]|[[omega].sub.n](x)|[D.sub.k](x)

if k = 4r + 3, r = 0, 1, 2, ...,

|[R.sub.[k,j].sup.B](x)|[less than or equal to][[|[B.sub.k]|]/[k!]]|[[omega].sub.n](x)|[D.sub.k](x),

if k = 2r, r = 1, 2,. ...

Proof. The proof follows from the above theorem and the following properties of Bernoulli polynomials:

[parallel][B.sub.k][[parallel].sub.[infinity],1] = [max[gamma]]|[B.sub.k](t)|[less than or equal to]|[B.sub.k]|,k = 2r,r = 1,2, ...

[parallel][B.sub.k][[parallel].sub.[infinity],1][less than or equal to][[2k!]/[(2[pi]).sup.k]][1/[1 - [2.sup.[1 - k]]]],k = 4r + 3,4 = 0,1,2, ....

Lemma 19. Let [DELTA] = {[x.sub.0] = a < [x.sub.1] < ... < [x.sub.n] = b} be a given uniform subdivision of the interval [a, b], i.e. [x.sub.i] = [x.sub.0] + ih, h = (b-a)/n, i = 0, 1, 2, ..., n. If x [member of] ([x.sub.j]-1, [x.sub.j]), for some j [member of] {1, 2, ..., n}, then

|[[omega].sub.n](x)|[less than or equal to]j!(n - j + 1)![h.sup.[n + 1]],(43)

[C.sub.k](x)[less than or equal to][[2.sup.n]/[n!]][[lcub[1/2][n + 1 + 1|n - 2j + 1|]rcub].sup.k][h.sup.[k - n]],(44)

and

[C.sub.k](x)|[[omega].sub.n](x)|[less than or equal to][[alpha].sub.jnk][[n - j + 1]/n][[[2.sup.n][[(b - a)].sup.[k + 1]]]/(nj)],(45)

where

[[alpha].sub.jnk] = [[1/[2n](n + 1 + |2j - n - 1|)].sup.k].(46)

This lemma is proved in (11).

Remark 20. Note that

[alpha].sub.jnk] [less than or equal to] 1

and [[alpha].sub.jnk] = 1 if and only if j = 1 or j = n. If we choose x [member of] [[x.sub.j],[x.sub.j]+1], j = 0, 1, ..., n-1, then we get the factor (j + 1)/n instead of the factor (n-j + 1)/n in (45).

Theorem 21. Under the assumptions of Lemma 19 and Theorem 15 we have

|[R.sub.[k,j].sup.B](x)|[less than or equal to][[|[B.sub.2r]|[[alpha].sub.jnk]]/[k!]][[1 - [2.sup.[ - 2r]]]/r][[j + 1]/n][[[2.sup.n][[(b - a)].sup.[k + 1]]]/(nj)]([[GAMMA].sub.[k + 1]] - [[gamma].sub.[k + 1]]),

if k is odd, k = 2r - 1, r [greater than or equal to] 1 and

|[R.sub.[k,j].sup.B](x)|[less than or equal to][[[[LAMBDA].sub.k][[alpha].sub.jnk]]/[(k + 1)]][[n - j]/n][[[2.sup.n][[(b - a)].sup.[k + 1]]]/(nj)]([[GAMMA].sub.[k + 1]] - [[gamma].sub.[k + 1]]),

if k is even and where [[LAMBDA].sub.k] is defined by (38).

Proof. The proof follows immediately from Corollary 16 and Lemma 19.

Remark 22. Here we also have an additional estimate

|[R.sub.[k,j].sup.B](x)|[less than or equal to][[[[GAMMA].sub.[k + 1]] - [[gamma].sub.[K + 1]]]/[2[square root of [(2k)!]]]][square root of [|[B.sub.2k]|]][[alpha].sub.jnk][[n - j + 1]/n][[[2.sup.n][[(b - a)].sup.[k + 1]]]/(nj)].

We now consider the interpolation formula given in Corollary 6.

First we recall some properties of the Euler polynomials. The Euler polynomial sare defined by the relation

[[2[e.sup.xt]]/[[e.sup.t] + 1]] = [[[infinity]].summation over (k = 0)][E.sub.k](x)[[t.sup.k]/[k!]],|t|<[pi],

such that

[E.sub.0](x) = 1,[E.sub.1](x) = x - [1/2],[E.sub.2](x) = [x.sup.2] - x, ...

The numbers [E.sub.k] = [2.sup.k] [E.sub.k](1/2) are Euler numbers; [E.sub.0] = 1,[E.sub.2] = -1, [E.sub.4] = 5, [E.sub.2k+1] = 0, k = 1, 2, ... .

We have the properties

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Further properties of these polynomials can be found in (1).

We specially have

[[integral].sub.0.sup.1]|[E.sub.k](t)|dt = 2|[[integral].sub.0.sup.[1/2]][E.sub.k](t)dt| = 2|[[[E.sub.k] + 1(1/2) - [E.sub.[k + 1]](0)]/[k + 1]] = 2|[[[2.sup.[ - k - 1]][E.sub.[k + 1]] - [E.sub.[k + 1]](0)]/[k + 1]]|. (47)

We also introduce the notation

[[theta].sub.k] = 2|[2.sup.[ - k - 1]][E.sub.[k + 1]] - [E.sub.[k + 1]](0)|. (48)

Theorem 23. Under the assumptions of Corollary 6 we have

|[R.sub.[k,j].sup.E](x)|[less than or equal to][[[theta].sub.k]/[(k + 1)!]][G.sub.j]([f.sup.[(k + 1)]])[C.sub.k](x)|[[omega].sub.n](x)|,

where [[theta].sub.k] is defined by (48).

Proof. We substitute the polynomials (10) in Theorem 10 and note that

|[[integral].sub.[x.sub.i].sup.x]|[E.sub.k]([t - [x.sub.i]]/[x - [x.sub.i]])|dt| = |x - [x.sub.i]|[[integral].sub.0.sup.1]|[E.sub.k](t)|dt = [[[theta].sub.k]/[k + 1]]|x - [x.sub.i]|. (49)

Theorem 24. Let the assumptions of Corollary 6 hold. If [[gamma].sub.k+1], [[GAMMA].sub.k+1] are real numbers such that [[gamma].sub.k+1] [less than or equal to] [f.sup.(k+1](t) [less than or equal to] [[GAMMA].sub.k+1], t [member of] [a, b], k = 0, 1, ..., n-1, then

|[R.sub.[k,j].sup.E](x)|[less than or equal to][[[theta].sub.k]/[2(k + 1)!]]([[GAMMA].sub.[k + 1]] - [[gamma].sub.[k + 1]])|[[omega].sub.n](x)|[C.sub.k](x),

where [[omega].sub.n] (x) is defined by (5). We also have

|[R.sub.[k,j].sup.E](x)|[less than or equal to][[[4.sup.[ - k/2]]|[E.sub.k]|]/[k!]]|[[omega].sub.n](x)|[F.sub.k](x),

|[R.sub.[k,j].sup.E](x)|[less than or equal to][[[4.sup.[ - k/2]]|[E.sub.k]|]/[k!]]|[[omega].sub.n](x)|[D.sub.k](x),

if k = 4r, r = 1, 2, ..., and

|[R.sub.[k,j].sup.E](x)|[less than or equal to][4/[[pi].sup.[k + 1]]](1 + 1/[[2.sup.[k + 1]] - 2])|[[omega].sub.n](x)|[F.sub.k](x),

|[R.sub.[k,j].sup.E](x)|[less than or equal to][4/[[pi].sup.[k + 1]]](1 + 1/[[2.sup.[k + 1]] - 2])|[[omega].sub.n](x)|[D.sub.k](x),

if k = 4r-1, r = 1, 2,. ...

Proof. We substitute the polynomials (10) in Theorem 11 and use the following properties of Euler polynomials:

[max[t[member of][0,1]]]|[E.sub.k](t)|[less than or equal to][4.sup.[ - k/2]]|[E.sub.k]|,k = 4r,r = 1,2, ...,

[max[t[member of][0,1]]]|[E.sub.k](t)|[less than or equal to][[4k!]/[[pi].sup.[k + 1]]](1 + 1/[[2.sup.[k + 1]] - 2]),k = 4r - 1,r = 1,2,. ...

Corollary 25. Under the assumptions of Lemma 19 and Theorem 24 we have

|[R.sub.[k,j].sup.E](x)|[less than or equal to][[[[theta].sub.k][[alpha].sub.jnk]]/[(k + 1)!]][[n - j + 1]/n][[[2.sup.[n - 1]][[(b - a)].sup.[k + 1]]]/(nj)]([[GAMMA].sub.[k + 1]] - [[gamma].sub.[k + 1]]).

Proof. The proof follows immediately from Theorem 24 and Lemma 19.

Remark 26. Some of the above inequalities ( for this choice of polynomials [P.sub.k](t)) are improvements of corresponding error inequalities obtained in (11).

Finally, we consider the interpolation formula given in Corollary 9.

Theorem 27. Under the assumptions of Corollary 9 we have

|[R.sub.[k,j].sup.T](x)|[less than or equal to][[[G.sub.j]([f.sup.[(k + 1)]])]/[(k + 1)!]][C.sub.k](x)|[[omega].sub.n](x)|.

Proof. We substitute the polynomials (23) in Theorem 10.

Theorem 28. Let the assumptions of Corollary 9 hold. If [[gamma].sub.k+1], [[GAMMA].sub.k+1] are real numbers such that [[gamma].sub.k+1] [less than or equal to] [f.sup.k+1](t) [less than or equal to] [[GAMMA].sub.k+1], t [member of] [a, b], k = 0, 1, ..., n-1, then

|[R.sub.[k,j].sup.T](x)|[less than or equal to][[[[GAMMA].sub.[k + 1]] - [[gamma].sub.[k + 1]]]/[2(k + 1)!]][C.sub.k](x)|[[omega].sub.n](x)|,

where [[omega].sub.n](x) is defined by (5). We also have

|[R.sub.[k,j].sup.T](x)|[less than or equal to][[|[[omega].sub.n](x)|]/[k!]][F.sub.k](x)

and

|[R.sub.[k,j].sup.T](x)|[less than or equal to][[|[[omega].sub.n](x)|]/[k!]][D.sub.k](x)

Proof. We substitute the polynomials (23) in Theorem 11.

Corollary 29. Under the assumptions of Lemma 19 and Theorem 28 we have

|[R.sub.[k,j].sup.T](x)[less than or equal to][[[alpha].sub.jnk]/[(k + 1)!]][[n - j + 1]/n][[[2.sup.[n - 1]][[(b - a)].sup.[k + 1]]]/(nj)]([[GAMMA].sub.[k + 1]] - [[gamma].sub.[k + 1]]).

Proof. The proof follows immediately from Theorem 28 and Lemma 19.

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Josip Pecaric Faculty of Textile Technology, University of Zagreb Pierottijeva 6, 10000 Zagreb, Croatia and

Nenad Ujevic [dagger] Department of Mathematics, University of Split Teslina 12/III, 21000 Split, Croatia

Received October 16, 2008, Accepted Augest 12, 2009.

* 2000 Mathematics Subject Classification. Primary 65D05, 41A80, 26D10.

[dagger] Corresponding author. E-mail:ujevic@pmfst.hr