On the form of the remainder in some approximation formulas of analysis

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Tiberiu Popoviciu

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T. Popoviciu, Asupra formei restului în unele formule de aproximare ale analizei, Lucrările ses. generale ştiinţifice a Acad. R.P.R., 2-12 iunie 1950, pp. 183-186 (in Romanian)

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1950 b -187 -Popoviciu- Lucr. Ses. Gen. St. Acad. RPR – Asupra formei restului in unele formule de aproximatie ale analizei

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1950 b -187 -Popoviciu- Work. Sess. Gen. St. Acad. RPR - On the form of the remainder in some formulas of a

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ON THE FORM OF THE REMAINDER IN SOME APPROXIMATION FORMULAS OF ANALYSIS

OFPROFESSOR TIBERIU POPOVICIUCORRESPONDING MEMBER OF ACAJOEMEI RRR

Communication presented at the meeting of March 3, 1950.

I. The remainder in the formulas for derivation and approximate integration is usually presented in the form of a linear functional

R [t]

The field of this functional, in the simplest and most important cases in practice, is the field of continuous functions, possibly of differentiable functions, or adding continuous derivatives up to a certain given order inclusive, over a finite and closed interval.

[a, b]

.

If the approximation has the character of a polynomial approximation, the degree of accuracy

n

of the respective formula is well determined by the property that

\begin{matrix} (I) & R [I] = R [x] = \dots = R [x^{n}] = 0, R [x^{n + 1}] \neq 0 \end{matrix}

We can, for simplicity, limit ourselves to the case

R [

I

] = 0

So we have

n ⩾ 0

The respective approximation formula is therefore exact (without remainder) for any polynomial of degree

n

.
2. For practical applications of approximation formulas it is important to express the remainder in a convenient form. This is usually done in the case of

R [f]

considered above, using the derivative

f^{n + 1} (x)

of the order

n + 1

of the function

f (x)

, assuming of course that this derivative exists, possibly that it is continuous.

The problem of the remainder has been studied by a large number of authors. We recall here first of all

A

.

A

. Marcov (I) and we point out the research of GD Birkhoff [2], G. Kowalewschi [3] and J. Radon [4]. The problem was then taken up again, in the light of new research on linear functionals, by Ez. Y a. Remez in an important memoir [5].

Our research on higher-order convex functions naturally leads to a new form of the remainder, a form which is likely to unify the results obtained so far, at the same time putting some of these results in
a more general form and explaining, we believe, the structure of this remainder more deeply.
3. The well-known properties of divided differences indicate that, in the case of the linear functional

R [f]

, checking the conditions ( I ), let's look for this functional in the form

\begin{matrix} (2) & R [f] = K . [x_{1}, x_{2}, \dots, x_{n + 2}; f] \end{matrix}

where

K

is independent of the function

f (x)

and

x_{1}, x_{2}, \dots, x_{n + 2}

are

n + 2

points in interval1

[a, b]

(points that generally depend on the function

f (x)

).

REST

R [f]

does not always have this form. In the case where this is possible with a

K

different from zero and with the points

x_{1}, x_{2}, \dots, x_{n + 2}

separate for anything

f (x)

, we will say that the rest

R [f]

has the classical form. The name is justified by the fact that in the important case of Taylor's formula or more generally of Lagrange's interpolation formula, the remainder has this form.

It is clear that for a linear functional

R [f]

, non-identical null, the classical form can only occur for a single

n

, equal to the degree of accuracy, and that then

K = R [x^{n + 1}]

.

To recognize the classical form, we have the following theorem:
For the remainder R [f] with accuracy degree n to have the classical form, it is necessary and sufficient that

R [f] \neq 0

, for any function

f (x)

convex of order n.

The condition is obviously necessary. It is sufficient as follows from a further result of ours [6].

From the above theorem, various simpler criteria result which allow in certain cases to recognize the classical form of the remainder. Such criteria result directly from the research of the authors cited above and directly from the theory of higher-order convex functions [6]. It is immediately seen that the problem of the classical form of the remainder returns to the problem of linear inequalities verified by higher-order convex functions. It is worth noting that the problem of the classical form of the remainder, for the most important formulas of derivation and numerical integration, returns to the problem of such inequalities on a finite number of points.

We add that important approximation formulas such as AA Marcov's approximate derivative formulas, Cotes' and Gauss' numerical integration formulas, as well as many formulas of this nature discovered and used by SE Mike1adze [7], [8], fall into the classical form of the remainder.
4. If the remainder

R [f]

does not have the classical form, its expression using divided differences is more complicated. In order not to complicate things unnecessarily here, let us assume that

R [f]

is defined in the field of functions with

a (n + I)

,

a

bounded divided difference (more precisely: the field of definition of

R [f]

contains these functions). Either

(C)

function class

f (x)

having the difference divided by the order

n + 2

between 0 and I exclusively, so

0 < [x_{1}, x_{2}, \dots, x_{n + 2}; f] < I

whatever the distinct points

x_{1}, x_{2}, \dots, x_{n + 2}

FROM

[a, b]

and either

A = sup R [∤]

f (x) \in (C)

.

A

is finite and, without loss of generality, we can assume

A ≧ 0

. Then I have

\begin{aligned} inf R [f] = R [x^{n + 1}] - A = - B ≦ 0 . \\ f (x) \in (C) \end{aligned}

It is then proven that we have

\begin{matrix} (3) & R [f] = A [x_{1}, x_{2}, \dots, x_{n + 2}; f] - B [x_{1}^{'}, x_{2}^{'}, \dots, x_{n + 2}^{'}; f] \end{matrix}

where the points

x_{i}, x_{i}^{'}

I am in

[a, b]

We have the following theorem:
If the upper boundary A is not touched by any of the functions

f (x)

from the class

(C)

, then the formula

(3)

is true for points

x_{i}

distinct and for points

x^{'}_{l}

distinct.

The theory of higher-order convex functions also allows us to recognize, in cases important for practice, when this occurs.

For the actual determination of the upper edge

A

, the results of E. Remez [5] can be used.

cases

B = 0

, and

A = 0

corresponding to the classical form of the remainder. The above theory is valid for any

n

.
5. The restrictions imposed above exclude from our considerations certain linear functionals

R [f]

, such as for example

R [f] = f^{n + 1} (0)

For such cases, formula (3) still remains valid, but without the restriction that the points

x_{l}

, respectively the points

x^{'}_{i}

, must be distinct.

It can also be observed that, without restriction imposed on the upper boundary

A

and if

ε

is a positive number, the formula

R [f] = (A + ε) [x_{1}, x_{2}, \dots, x_{n + 2}; f] - (B + ε) [x^{'}_{1}, x^{'}_{2}, \dots, x^{'}_{n + 2}; f]

is valid with points

x_{i}

, respectively with points

x_{i}^{'}

distinct. It follows, in particular, that if

f (x)

has a difference divided by the order

n + 1

limited in absolute value by the number

M

, we have

| R [f] | ≦ (A + B) M .

The proofs and applications of the previous results will be presented in a detailed paper that is currently being prepared.

SUMMARY

The remainder in the derivation and numerical integration formulas is represented in the simplest cases under the form of a linear functional

R [f]

The theory of higher-order convex functions allows us to express this remainder in the form (3), when

A, B

, independent of the function

f (x)

Cases

A = 0

or

B = 0

are especially interesting, and we then have the classical form of the remainder. For әто to take place, it is necessary and sufficient that in hypothesis (1) there would be

R [f] = 0

for any convex function of order

n f (x)

. In (2) points

x_{i}

are assumed to be excellent.

DRAWING

The remainder in the numerical derivation and integration formulas is presented, in the simplest cases, in the form of a linear functional

R [f]

. The theory of higher-order convex functions allows us to express this remainder in the form (3),

A, B

being independent of the function

f (x)

. The case

A = 0

, or

B = 0

, is particularly interesting; we then have the classical form of the remainder. The necessary and sufficient condition for this to be so is that, under the assumption (

I

), we have

R [f] \neq 0