On a quadrature formula of S. Golab and C. Olech

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Tiberiu Popoviciu

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Tiberiu Popoviciu
(Institutul de Calcul)

Original title (in French)

Sur une formule de quadrature de S. Golab et C. Olech

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T. Popoviciu, Sur une formule de quadrature de S. Golab et C. Olech, Demonstratio Math., 6 (1973), pp. 771-789 (1974) (in French)

(Collection of articles dedicated to Stanislaw Golab on his 70th birthday)

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1973 a -Popoviciu- Demonstratio Math. – Sur une formule de quadrature de S. Golab et C. Olech

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Demonstratio Math.

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Warsaw University of Technology

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Tiberiu Popoviciu

ON A SQUADRATION FORMULA BY S. GOŁAB AND C. OLECH

Dedicated to MS Gołąb on the occasion of his $70^{ième}$ birthday

S. Gołąb and C. Olech demonstrated [1] that there exists one and only one number strictly between 0 and 1 $(0 < (4 < 1)$ , so that the quadrature formula
(1) $\int_{0}^{1} f (x) d x \cdot = λ_{0} f (0) + λ_{1} f (0) + λ_{2} f (1) + R (f)$
be exact for the functions ( $x^{0} =$ ) $1, x^{p}, x^{q}, x^{r}$ , Or $p, q$ , $r$ are any positive numbers such that $p < q < r$ The fact that formula (1) is exact for functions 1, $x^{p}$ , $x^{q}, x^{r}$ means that
(2) $R (1) = R (x^{p}) = R (x^{q}) = R (x^{r}) = 0$ .

The coefficients

λ_{0}, λ_{1}, λ_{2}

are independent of the function

f

and can be easily calculated using the equalities (2). We thus obtain

λ_{0} + λ_{1} + λ_{2} = 1

and
(3a)

λ_{1} = \frac{q - p}{(p + 1) (q + 1) (ω^{p} - θ^{q})}

;
(3b)

λ_{2} = \frac{(p + 1) θ^{p} - (q + 1) θ^{q}}{(p + 1) (q + 1) (θ^{p} - (θ^{q})} .

Any real power of a positive number is a well-defined positive number. Other determinations of such a power will not be relevant to this discussion. We have

0^{σ} = 0

if 6 is positive. The function

x^{6}

is therefore defined, uniform, continuous and infinitely differentiable on the positive real axis, for any real exponent

σ^{'}

. For

σ = 0

this function reduces to the constant 1. If

σ > 0

the function

x

is defined and continuous for

x ⩾ 0

.

The number (10) is given by the equation ].

\begin{matrix} (4) & (r = q) (p + 1) O^{p} + (p - r) (q + 1) Q^{q} + \\ + (q - p) (r + 1) O^{r} = 0 \end{matrix}

which, as S. Goląb and C. Olech have shown [1], has exactly one and only one root in the open interval

] 0, 1 [

In the work cited from

S

Gołąb and C. Olech assume that the exponents

p, q, r

are (positive) integers. Following their proof, we can easily see that this restriction is not necessary. We will therefore only assume that

0 < p < q < r

.

Formula (1) is an interesting generalization of Simpson's formula; the latter is obtained by taking

p = 1; q = 2, r = 3

.

In this work we propose to study the remainder

R (1)

of formula (1), under well-defined assumptions made about the function

f

。

Beforehand, we will establish some results which, being of interest in themselves, can be used in the study of other problems, analogous to the one dealt with in this work.

2. Let us designate by

(5)

V (\begin{array}{ll} g_{1}, & g_{2}, \dots, g_{n} \\ x_{1}, & x_{2}, \dots, \end{array}) = {| g_{n} (x_{i}) |}_{i, t = 1, 2, \dots, n}

1st order determinant

n

values of the functions

g_{t}, t == 1, 2, \dots, n

on the points

x_{i}, i = 1, 2, \dots, n

In this determinant

g_{t} (x_{i})

is the element that is located in the i-th row and twi-th column.

The determinant (5) is obviously zero if the points

x_{\hat{1}}

or if the functions

g_{t}

are not distinct.

If the points

x_{i}

are not distinct, notation (5) will be used for a suitably modified determinant. This modification consists of replacing the lines corresponding to each group of points

x_{i}

confused by lines formed by the values of the functions

g_{t}

and their successive derivatives at these points. This implies, of course, the existence of the derivatives considered. More precisely, let

z_{1}, z_{2}, \dots, z_{m}

the distinct points with which coincide respectively

k_{1}, k_{2}, \dots, k_{1 n} (k_{1}, k_{2,} \dots, k_{m} ⩾ 1

) points

x_{i}

So for everything

i = 1, 2, \dots, m

, there is exactly

k_{i}

lines made up of the function values

g_{t}

and their

k_{i} - 1

first derivatives on the point

z_{j}

.

We continue to refer to by

\begin{matrix} (6) & V (\begin{array}{llll} g_{1}, & g_{2}, & \dots, & g_{n} \\ x_{1}, & x_{2}, & \dots, & x_{n} \end{array}) \end{matrix}

the determinant thus modified. But it is important to precede the succession of lines of the determinant thus defined, which is indeed of order

k_{1} + k_{2} + \dots + k_{m} = n

.

We will denote by (6) the determinant of order n whose

{(k_{1} + k_{2} + \dots + k_{i - 1} + j)}^{i emes}

lines

_{β}

for consecutive values 1, 2, ...,

k_{i}

of

j

, taken in this order, are the following
(7)

{\begin{cases} g_{1} (z_{i}) & g_{2} (z_{i}) & \dots & g_{n} (z_{i}) \\ g_{1}^{'} (z_{i}) & g_{2}^{'} (z_{i}) & \dots & g_{n}^{'} (z_{i}) \\ \cdot & \dots & \cdot & \dots \\ (k_{i} - 1) & (z_{i}) & g_{2} (k_{i} - 1) & (z_{i}) \end{cases} \dots \cdot g_{n}^{(k_{i} - 1)} (z_{i})

.

The sum

k_{1} + k_{2} + \dots + k_{i - 1}

is replaced by 0 for

i = 1

and the accents denote successive derivations.

The lines (7) are consecutive lines in the determinant (6). With this convention the succession of columns and lines is clearly specified in the determinant (6).

Of course, the modified determinant (6) is well-defined even if the points

z_{1}, z_{2}, \dots, z_{m}

are not distinct, but then it is obviously equal to 0.

Everything comes down, we sum up, to asking
(8)

x_{k_{1} + k_{2} + \dots + k_{i - 1} + j} = z_{i}, j = 1, 2, \dots, k_{i}, i = 1, 2, \dots, m

.

It is important to note that, in general, the lines of the determinant cannot be subjected to any permutation, since the order of these lines has been fixed in advance. By this we mean that, to take an example, while the determinant

V (\begin{array}{lll} E_{1}, & E_{2}, & E_{3} \\ z_{1}, & z_{1}, & z_{2} \end{array})

is always defined, the symbol

V (\begin{array}{lll} g_{1}, & g_{2}, & g_{3} \\ z_{1}, & z_{2}, & z_{1} \end{array})

means nothing if

z_{1} \neq z_{2}

.

On the contrary, we can subject the lines of the determinant to a permutation by group of points combined; which amounts, in short, to a permutation of the points

z_{i}

More precisely, either

v_{1}, v_{2}, \dots, v_{n}

a permutation of indices

1, 2, \dots, n

Let us then assume, to simplify the notation.

x_{i}^{'} = x_{v_{i}}, i = 1, 2, \dots, n

The determinant (6), with the corresponding rows permuted, is then

v (\begin{array}{llll} g_{1}, & g_{2}, & \dots, & g_{n} \\ x_{1}^{'}, & x_{2}^{'}, & \dots, & x_{n}^{'} \end{array})

1a succession of lines for a group of points

x_{i}^{'}

equals with a

z_{i}

being respected according to the rule described by table (7). This means that we only consider permutations

v_{1}, v_{2}, \dots, v_{n}

for which we have
(10)

x_{k_{1}^{'} + k_{2}^{'} + \dots + k_{i - 1}^{'} + j}^{'} = z_{i}^{'}, j = 1, 2, \dots, k_{i}^{'}, i = 1, 2, \dots, m

the sequel

z_{1}^{'}, z_{2}^{'}, \dots, z_{m}^{'}

being a permutation of the sequence

z_{1}, z_{2}, \dots, z_{m}

The value of the determinant (9) differs from that of (6) at most by the sign, according to formula (17) which will be established later.

Let us now point out the following special cases:

1^{\circ}

For the functions

g_{t} = x^{t - 1}, t = 1, 2, \dots, n

, the ending deus (5) reduces to the determinant of Vandermonde

V (x_{1}, x_{2}, \dots, x_{n})

numbers

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

So we have

\begin{aligned} (11) & V (\binom{1, x, \dots, x^{n - 1}}{x_{1}, x_{2}, \dots, x_{n}}) = V (x_{1}, x_{2}, \dots, x_{n}) = \\ = \prod_{i < j}^{1, 2, \dots, n} (x_{j} - x_{i}), (V (x_{1}) = 1) \end{aligned}

2^{0}

In the case where all the points

x_{i}

coincident (with x) the modified determinant (6) reduces to the Wronskian

W (g_{1}, g_{2}, \dots, g_{n}) (x)

functions

g_{t}^{*)}

We can therefore write
*) This is why we sometimes call the determinant (5) the Prewronskian of functions

E_{t}

.

V (\begin{array}{cccc} g_{1}, & g_{2}, & \dots, & g_{n} \\ x, & x, & \dots, & x \end{array}) = W (g_{1}, g_{2}, \dots, g_{n}) (x) .

We can also obtain the modified determinant (6) by a suitable limit. To avoid complicating matters, we will assume that all the derivatives involved exist and are continuous, at least in certain neighborhoods of the points $z_{1}$ .

So be it

n

distinct points

x_{j}^{(i)}, j = 1, 2, \dots, k_{j}

,

i = 1, 2, \dots, m_{, et considérons le déterminant D} d^{'}

order

n

(

= k_{1} + k_{2} + \dots + k_{m}

) of which the element of lå (

k_{1} + k_{2} + \dots + {+ \dots + k_{i - 1} + j)}^{i e m e}

line and the

t^{i e m e}

column is the usual divided difference

[x_{1}^{(i)}, x_{2}^{(i)}, \dots, x_{j}^{(i)}; g_{t}]

of the function

g_{t}

on the nodes

x_{1}^{(i)}, x_{2}^{(i)}, \dots, x_{j}^{(i)}

and where

j == 1, 2, \dots, k_{i}, t = 1, 2, \dots, n

If we observe that this divided difference tends towards

\frac{1}{(j - 1)!} g_{t}^{(j \dots 1)} (z_{1})

when the points

x_{v}^{(i)}, v = 1, 2, \dots, j

tend towards

z_{i}

, we see that the determinant

D

tends towards the modified determinant (6) divided by the number

\prod_{i = 1}^{m} (k_{i} - 1)

!! when

x_{V}^{(i)} \to z_{i}, v = 1, 2, \dots, k_{j}, i = 1, 2, \dots, m_{0}

We use the abbreviated notation

α!! = 1! 2! \dots α

! (

0!! = 1

).

Finally, if we multiply the determinant

D

by the product

\begin{matrix} (13) & \prod_{i = 1}^{m} v (x_{1}^{(i)}, x_{2}^{(i)}, \dots, x_{k_{i}}^{(i)}) \end{matrix}

and if we perform some elementary operations on the lines, we obtain the determinant
(14)

V (x_{1}^{(1)}, x_{2}^{(1)}, \dots, x_{k_{1}}^{(1)}, x_{1}^{(2)}, x_{2}^{(2)}, \dots, x_{k_{2}}^{(2)}, \dots, x_{1}^{(m)}, x_{2}^{(m)}, \dots, x_{k_{m}}^{(m)})

It follows that the determinant (6) is obtained by multiplying (14) pax

\prod_{i = 1}^{m} (k_{i} - 1) 11

, by dividing it by (13) and then making the points stretch

x_{j}^{(i)}

towards

z_{i}

, For

j == 1, 2, \dots, k_{i}, i = 1, 2, \dots, m

,

The concept of difference divided is well known. We

l^{"}

We will employ it in its general form as set forth in the memoir cited below concerning the study of the remainder of formula (1) [5]. The usual divided differences are those with respect to successive non-negative integer powers.

1, x, x^{2}, \dots

. of the variable.
4. As a first application, we will find the formula giving the value of the generalized Vandermonde determinant

\begin{aligned} (15) & V (1_{1}^{1}, x_{2}, \dots, x^{n - 1}) = V (x_{1}, x_{2}, \dots, x_{n}) = \\ = & \prod_{i = 1}^{1, 2, \dots, m} (k_{i} - 1)!! \prod_{i < j}^{1, 2, \dots, m} {(z_{j} - z_{i})}^{k_{i} k_{j}} \end{aligned}

using the notation (8). In particular, we have

V (\underset{m}{\underset{⏟}{x, x, \dots, x}}) = W (1, x, \dots, x^{n - 1}) (x) = (n - 1)!!

Returning to the value of the determinant (9), equal to (6) or its opposite, we can write

\begin{matrix} (16) & sg V (\begin{array}{ll} g_{1}, & g_{2}, \dots, g_{n} \\ x_{1}^{'}, & x_{2}^{'}, \dots, x_{n}^{'} \end{array}) = \\ = sg (V (x_{1}^{'}, x_{2}^{'}, \dots, x_{n}^{'}) V (\begin{array}{ll} g_{1}, & g_{2}, \dots, g_{n} \\ x_{1}, & x_{2}, \dots, \end{array})) \end{matrix}

x_{1}^{'}, x_{2}^{'}, \dots, x_{n}^{'}

being an admissible permutation of the sequence

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

assuming that

x_{1} ⩽ x_{2} ⩽ \dots ⩽ x_{n}

and where we employ the function

sg x = {\begin{aligned} 1, & pour \\ 0, & x > 0 \\ - 1, & pour \end{aligned} x = 0

which satisfies the functional equation

s g (x y) = s g x \cdot s g y

For

x

And

y

any real numbers.

Taking into account (15) and (16) we also obtain

\begin{matrix} (17) & sg V (\begin{array}{cccc} g_{1}, & g_{2}, & \dots, & g_{n} \\ x_{1}^{0}, & x_{2}^{0}, & \dots, & x_{n}^{'} \end{array}) = \end{matrix}

= sg (\prod_{j < j}^{1, 2, \dots, m} {(z_{j}^{'} - z_{i}^{'})}^{k_{i}^{'} k_{j}^{'}} v (\begin{array}{llll} g_{1}, & g_{2}, & \dots, & g_{n} \\ x_{1}, & x_{2}, & \dots, & x_{n} \end{array}))

using notation (10) and assuming that

z_{1} < z_{2} << \dots < z_{m}

.

Let us note in passing that if among the numbers

k_{1, 8} k_{2}

, ...,

k_{m}

he

y

a at most one which is odd, the determinant (9) does not depend on the admissible permutation

x_{1}^{'}

,

x_{2}^{'}, \dots, x_{n}^{'} d e

the sequel

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

5.
We will now establish a formula that reduces the calculation of a determinant of the form (6) of order n to the calculation of a determinant of the same form but of order n - 1.

Let's modify the lines of the determinant (6) as follows: Let's return to table (7). Leibniz's formula

\begin{matrix} g_{t}^{(u)} = {(\frac{g_{t}}{g_{1}} g_{1})}^{(u)} = {(\frac{g_{t}}{g_{1}})}^{(u)} g_{1} + \sum_{v = 1}^{u} (\binom{u}{v}) {(\frac{g_{t}}{g_{1}})}^{(u - v)} g_{1}^{(v)} \\ (v = 1, 2, \dots, k - 1, t = 1, 2, \dots, n) \end{matrix}

shows us that, without changing the value of the determinant (6),
the

k_{i} - 1

last lines of table (7) can be replaced respectively by

\begin{matrix} (18) & {\begin{cases} g_{1} h_{1}^{'} & g_{1} h_{2}^{'} & \dots & g_{1} h_{n}^{'} \\ g_{1} h_{1}^{''} & g_{1} h_{2}^{''} & \dots & g_{1} h_{n}^{''} \\ \dots & \dots & \dots & \dots \\ g_{1} h_{1}^{(k_{i} - 1)} & g_{1} h_{2}^{(k_{i} - 1)} & \dots & g_{1} h_{n}^{(k_{i} - 1)} \end{cases} \end{matrix}

Or

h_{t} = g_{t} / g_{1}, t = 1, 2, \dots, n

, the functions and derivatives of various orders being calculated at the point

z_{i}

. If

k_{i} == 1

No such modification is made to table (7). Note that the elements in the first column of table (18) are all zero.

These results were obtained based on the fact that a determinant does not change value when any linear combination of other rows is added to a row. After these operations, the rows that remained unchanged form the table.

\begin{matrix} (19) & {\begin{cases} g_{1} (z_{1}) & g_{2} (z_{1}) & \dots & g_{n} (z_{1}) \\ g_{1} (z_{2}) & g_{2} (z_{2}) & \dots & g_{n} (z_{2}) \\ \cdot & \cdot & \cdot & \cdot \\ g_{1} (z_{m}) & g_{2} (z_{m}) & \dots & g_{n} (z_{m}) \end{cases} \end{matrix}

These lines retain their positions in determinant (6). Still without changing the value of determinant (6), we can subtract (i-1).

^{i e ̀ m e}

line multiplied by

\frac{g_{1} (z_{i})}{g_{1} (z_{i - 1})}

of the

i^{i}

and this for the second line

i = 1, 2, \dots, m

The elements of the table (19) thus transformed are obtained by applying the (Newton-Leibniz) formula.

g_{t} (z_{i}) - g_{t} (z_{i - 1}) \frac{g_{1} (z_{i})}{g_{1} (z_{i - 1})} = g_{1} (z_{i}) \int_{z_{i - 1}}^{z_{i}} {(\frac{g_{t} (y)}{g_{1} (y)})}^{'} d y .

When

m = 1

Table (19) has only one row which is left unchanged.

After all these transformations, the elements of the first column of the determinant (6) all become zero, except for the first one, which is equal to

g_{1} (z_{1})

Finally, by taking out the lactour

\prod_{i = 1}^{m} {(g_{1} (z_{i}))}^{k_{i}} = \prod_{i = 1}^{n} g_{1} (x_{i})

and by expanding the determinant from the first column, we obtain the formula

\begin{matrix} (20) & V (\begin{array}{cccc} g_{1}, & g_{2}, & \dots, & g_{n} \\ x_{1}, & x_{2}, & \dots, & x_{n} \end{array}) = \end{matrix}

= (\prod_{i = 1}^{m} g_{1} (x_{i})) \int_{z_{1}}^{z_{2}} \int_{z_{2}}^{z_{3}} \dots \int_{z_{m - 1}}^{z_{m}} ψ (y_{1}, y_{2}, \dots, y_{m - 1}) d y_{1} d y_{2} \dots d y_{m - 1}

Or

\begin{matrix} (21) & ψ (y_{1}, y_{2}, \dots, y_{m - 1}) = \end{matrix}

= v (\underset{k_{1} - 1}{\underset{⏟}{{(g_{2} / g_{1})}^{\circ}, {(g_{3} / g_{1})}^{\circ}, \dots, {(g_{n} / g_{1})}^{\circ}}} (\underset{k_{2} - 1}{\underset{⏟}{z_{1}, z_{1}, \dots, z_{1}}}, y_{1}, z_{2}, z_{2}, \dots, z_{2}, y_{2}, \dots, y_{m - 1}, \underset{k_{m} - 1}{\underset{⏟}{z_{m} z_{m}, \dots, z_{m}}})

This is the formula
we wanted to establish. We also used the notations (8)

We assume, of course, that all the functions, derivatives, and integrals involved are meaningful. In particular, for example, that the function

g_{1}

does not cancel out, that the function

ψ (y_{1}, y_{2}, \dots, y_{m - 1})

is continuous, etc.

If

k_{1} = 1

the point

z_{i}

does not appear on the second member of (21). Finally, if

m = 1

Formula (20) is meaningless. In this case, it is replaced by the well-known formula [2]:
(22)

w (g_{1}, g_{2}, \dots, g_{n}) = g_{1}^{n} w ({(\frac{g_{2}}{g_{1}})}^{'}, {(\frac{g_{3}}{g_{1}})}^{'}, \dots, {(\frac{g_{n}}{g_{1}})}^{'})

.

Formula (20) should therefore be regarded as a generalization of the latter formula (22).
6. All the assumptions concerning the continuity, differentiability, integrability, etc., of the functions involved are verified if we apply the preceding results to the functions

g_{t} (x) = x^{σ_{t}}, t = 1, 2, \dots, n

defined for

x > 0

, THE

σ_{t}

being any given real numbers. If we note that, in this case,

{(\frac{g_{t} (x)}{g_{1} (x)})}^{'} = (σ_{t} - σ_{1}) x^{σ_{t} - σ_{1} - 1}, t = 1, 2, \dots, n,

formula (20) becomes (

m > 1

)

\begin{aligned} (23) & = (σ_{2} - σ_{1}) (σ_{3} - σ_{1}) \dots (σ_{n}, \dots, x_{n} \\ \times \int_{z_{1}}^{z_{2}} \int_{z_{2}}^{z_{3}} \dots \int_{z_{m - 1}}^{z_{m}} Φ {(x_{1}, x_{2} \dots x_{n})}^{σ_{1}} \times \end{aligned}

Or

\begin{matrix} Φ (y_{1}, y_{2}, \dots, y_{m - 1}) = \\ = x^{σ_{2} - σ_{1} - 1}, x^{σ_{3} - σ_{1} - 1}, \dots, x^{σ_{n} - σ_{1} - 1} \\ = V (\underset{k_{1} - 1}{\underset{⏟}{z_{1}, z_{1}, \dots, z_{1}}}, y_{1}, \underset{k_{2} - 1}{\underset{⏟}{z_{2}, z_{2}, \dots, z_{2}}}, y_{2}, \dots, y_{m - 1}, \underset{k_{m} - 1}{\underset{⏟}{z_{m}, z_{m}, \dots, z_{m}}}) \end{matrix}

From this formula we deduce
Theorem 1. With the previous notation, if

0 < z_{1} < z_{2} < \dots < z_{m}

And

σ_{1} < σ_{2} < \dots < σ_{n}

, the determinant

\begin{matrix} (24) & v (\begin{array}{llll} x^{σ_{1}}, & x^{σ_{2}}, & \dots, & x^{σ_{n}} \\ x_{1}, & x_{2}, & \dots, & x_{n} \end{array}) \end{matrix}

is positive.

The demonstration presents no difficulties and is principally based on formula (23). We can proceed by complete induction. Let us first note that the property is via for

n = 1

because then the determinant (24) is reduced &

x_{1}^{6}

which is indeed a positive number. Let's also remember that for

n = 1

(

n

(any) the property is true guisqu alons on a

\begin{aligned} (25) & V (\binom{x_{1}^{σ_{1}}, x_{2}^{σ_{2}}, \dots, x^{σ_{n}}}{x_{1}, x_{1}, 000 x_{1}}) = W (x^{σ_{1}}, x^{σ_{2}}, \dots 0, x_{n}^{σ_{n}}) (x_{1}) = \\ = V (σ_{1}, σ_{2}, \dots, σ_{n}) x_{1}^{σ_{1}} + σ_{2} + \dots + σ_{n} - \frac{a (n - 1)}{2} \end{aligned}

Let us now assume that the property is true for the order determinant

n - 1 (n - 1)

and demonstrated for the order determinant

n

We can assume m

>↑

The property then results from formula (23) in which we have

z_{1} ⩽ y_{1} ⩽ z_{2} ⩽ \dots ⩽ y_{m - 1} ⩽ z_{m}

and the function to be integrated is continuous over its entire domain of definition

[z_{1}, z_{2}] \times [z_{2}, z_{3}] \times \dots x [z_{m - 1}, z_{m}]

and is (by hypothesis) positive at every internal point of this domain.

If we now take into account the general formula (16), we deduce the

Corollary 1. If the numbers

x_{i}, i = 1, 2, \dots, n

are positive, we have

\begin{matrix} (26) & sg V (\binom{x^{σ_{1}}, x^{σ_{2}}, \dots, x^{σ_{n}}}{x_{1}, x_{2}, \dots, x_{n}}) = \\ = sg (V (x_{1}, x_{2}, \dots, x_{n}) V^{π} (σ_{1}, σ_{2}, \dots, σ_{n})) \end{matrix}

THE

z_{i}

and the

k_{i}

being always determined by formulas (8).

In the second member

V (x_{1}, x_{2}, \dots, x_{n})

is the generalized Vandermonde determinant (15), but

V^{2} (σ_{1}, σ_{2}, \dots, σ_{n})

is the usual Vandermonde determinant of numbers

σ_{1}, σ_{2}, \dots, σ_{n}

, different or not, given by formula (11). Formula (26) is also true in the case where the

σ_{t}

are not all distinct, since then both members are equal to 0.

The special case

m = n

, therefore the case where the points

x_{i}

.

i = 1, 2, \dots, n

, are distinct, has also been dealt with by us in an earlier work [3].
7. We can now return to the study of the remainder of the quadrature formula (1). For this, we will use our results concerning generalized convex functions and the notion of a linear functional of simple form, presented in our work cited [5]. The reader is advised to refer to this work for the justification of all the statements that follow.

Formula (1) can be obtained by replacing the function f with the (generalized) Lagrange-Hermite type polynomial

L (x) = c_{0} + c_{1} x^{p} + c_{2} x^{q} + c_{3} x^{x}

whose coefficients

c_{0}, c_{1}, c_{2}, c_{3}

are well determined by the interpolatory conditions
(27)

L (0) = f (0), L^{'} (

四

) = f (

四),

L^{'} (0) = f^{'} (\oplus), L (1) = f (1)

.

Numbers

p, q, r

and (ii) have the meanings given in rr. 1.

As a result, the rest

R (f)

is given by the formula

\begin{matrix} (28) & R (f) = \int_{0}^{1} (f (x) - I (x)) d x \end{matrix}

But it's easy to point out the difference

f (x) = L (x)

in a remarkable form to

I

using the generalized divided difference
(29)

\begin{matrix} [x_{1}, x_{2}, x_{3}, x_{4}, x_{5}; f] = \\ = v (\binom{1, x^{p}, x^{q}, x^{r}, f}{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}}) : v (\binom{1, x^{p}, x^{q}, x^{r}, x^{s}}{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}}) \end{matrix}

Or

s

is a positive number

> r (> q > p)

We see that
, by virtue of corollary 1, this divided difference exists regardless of the points

x_{1}, x_{2}, x_{3}, x_{4}, x_{5}

, non-negative, but of which at most one is equal to

O

The two determinants of the second member of (29) are taken in the generalized sense (9) (with the

z_{j}

distinct). Note that if

x_{1} = 0

and the points

x_{2}, x_{3}, x_{4}, x_{5}

are positive, we have #)

V (\binom{1, x^{p}, x^{q}, x^{r}, x^{s}}{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}}) = V (\binom{x^{p}, x^{q}, x^{r}, x^{s}}{x_{2}, x_{3}, x_{4}, x_{5}})

We then have

\begin{matrix} (30) & f (x) - L (x) = \frac{V (\binom{1, x^{p}, x^{q}, x^{r}, x^{s}}{0, 0, 0, 1, x})}{V (\binom{1, x^{p}, x^{q}, x^{r}}{0, 0, 0, 1})} [0, 0, 0, 1, x; f] \end{matrix}

assuming that

x

be positive and different from

Θ

and 1. In our example

x

alone is always a group of points

z_{i}

In this way the determinant

V (\binom{1, x^{p}, x^{q}, x^{x}, x^{s}}{0, Θ, Θ, 1, x})

is zero if

x

coincides with 0, or 1, and then the difference divided

[0, (4, 0, 1, x_{9} f] n

'is not defined. This difficulty can easily be remedied, but it is unnecessary to do so in this work. For

x = 0, Θ

or 1 the second member of (30) does not make sense, but then the polynomial values

I

are given by (27).
8. The coefficient of the divided difference of the right-hand side of (30) is, by virtue of corollary 3, negative at every point of

[0, 1]

, outside of 0, (1) and 1.

Let us now recall that the function

f

is said to be convex with respect to the sequence of functions 1,

x^{p}, x^{q}, x^{r}, x^{s}

on the interval [0,1] if the divided difference (29) is

> 0

for any group of 5 distinct points

x_{1}, x_{2}, x_{3}, x_{4}, x_{5}

In this case, the divided difference (29) is still positive if the points

x_{1}, x_{2}, x_{3}, x_{4}, x_{5}

are not all confused, at least in the case where this difference divides exists. In the case (the only one that interests us here) where the function

f

is convex we have

[0, (1), (1), 1, x : f] > 0

For

x

different

0, (4)

and 1. If

t \in C [0, 1]

, therefore if the function

f

is continuous on

[0, 1]

, it follows from formulas (28) and (30) that if, in addition, the function is convex with respect to the sequence

1, x^{p}, x^{q}, x^{1}, x^{5}

We have

R (f) \neq 0 (p l u s

Exactly

R (f) < 0)

.

But

R (f)

is a linear (additive and homogeneous) functional defined on

C [0, 1]

It therefore follows from the study we have carried out on linear functionals of the simple form (see again work [5]) that

Theorem 2. If

f \in C [0, 1]

, the rest

R (f)

of the quadrature formula (1) is of the form

\begin{matrix} (31) & R (f) = R (x^{s}) [ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}, ξ_{5}; f] \end{matrix}

Or

ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}, ξ_{5}

are 5 distinct points of the open interval

[0, 1]

(generally dependent on the function)

f

) and the divided difference of the second member is defined by formula (29).

The coefficient

R (x^{s}) = \int_{0}^{1} x^{s} d x - λ_{1} (1)^{s} - λ_{2}

The value of formula (31) can be calculated using the values (3a) and (3b) of the coefficients

λ_{1}, λ_{2}

from formula (1). We obtain

(p + 1) (q - s) 0^{p} + (q + 1) (s - p) 0^{q}

\begin{matrix} (32) & R (x^{s}) = \frac{+ (s + 1) (p - q) (1)^{s}}{(p + 1) (q + 1) (s + 1) (1 (m)} \end{matrix}

where (10) Є

] 0, 1 [

satisfied &

1^{\circ}

equation (4).
Apparently, in the proof of Theorem 2, in addition to the continuity of the function f, its differentiability also comes into play, at least with respect to point (ii). But this theorem remains true without this last restriction. This results, on the one hand, from the fact that the derivative of the function I does not appear in formula (1) and, on the other hand, from the fact that a function is convex with respect to the sequence

1, x^{p}

,

x^{q}, x^{r}, x^{s}

on

1^{intervalle} [0, 1]

eat necessarily (continuous and) differentiable on

] 0, 1 [

We established this latter property previously [4].
9. Formula (31) is inconvenient for delimiting the remainder

R (I)

But, it also follows from our work cited [5], that if the function

i

has a deximate

4^{ieme}

on

1^{\circ}

interm valle ouvert ] 0,1 [ one can in (31) produce the points

ξ_{1} ξ_{2, 0} ξ_{3,} ξ_{4, 0} ξ_{5}

equal to a child point

ξ \in] 0_{0} 1 [

。

Formulas (12), (22) and (25) give us

V (\binom{1, x^{p}, x^{q}, x^{r}, l}{ξ, ξ, ξ, ξ, ξ}) = p q x W (x^{p - 1}, x^{q - 1}, x^{r - 1}, l^{'}) (ξ)

V (\binom{1, x^{p}, x^{q}, x^{r}, x^{s}}{ξ, ξ, ξ, ξ, ξ}) = ξ^{p + q + x + s - 10} p q r s V (p, q, r, s)

assuming

ξ \in] 0, 1 [

By doing the calculations, we will find

\begin{matrix} W (x^{p - 1}, x^{q - 1}, x^{r - 1}, f^{'}) (ξ) = \\ = ξ^{p + q + r - 9} V (p, q, r) {ξ^{3} f^{I V} (ξ) - (p + q + r - 6) ξ^{2} f^{'''} (ξ) + \\ + [(p - 1) (q - 2) + (p - 1) (r - 1) + (q - 2) (r - 2)] ξ f^{''} (ξ) + \\ - (p - 1) (q - 1) (r - 1) f^{'} (ξ)} \end{matrix}

and we therefore deduce
Theorem 3. If the function

f \in C [0, 1]

has a fourth derivative on

] 0, 1 [

, the rest

R (f)

of formula (1) is given by

\begin{aligned} R (f) = \frac{R {(x^{s})}^{1 - s}}{s (s - p) (s - q) (s - r)} {ξ^{3} f^{' v} (ξ) \bar{\mp} \\ - (p + q + r - 6) ξ^{2} f^{'''} (ξ) + \\ + [(p - 1) (q - 2) + (p - 1) (r - 1) + (q - 2) (r - 2)] ξ f^{''} (ξ) + \\ - (p - 1) (q - 1) (r - 1) f^{'} (ξ)} \end{aligned}

Or

\dot{ε}

is a point in the open interval ]0,1[ (generally depending on the function .f).
10. We will finish with two examples.

E x

Example 1. From Simpson's formula

(p = 1, q = 2

,

r = 3

), the remainder is given by the formula

R (f) = \frac{\frac{1}{s + 1} - \frac{1}{3 \cdot 2^{s - 1}} - \frac{1}{6}}{s (s - 1) (s - 2) (s - 3)} ξ^{4 - s} f (ξ) (ξ \in] 0, 1 [)

If we take

s = 4

we find the rest well known

- \frac{1}{2880} f^{IV} (ξ)

By taking

s = \frac{7}{2}

respectively

s = 5

we find the remainder respectively in the form

- \frac{4 (\sqrt{2} - 1)^{2}}{945 \cdot \sqrt{2}} \sqrt{ξ} f^{''} (ξ), - \frac{1}{5760} \cdot \frac{f^{' V} (ξ)}{ξ} (ξ \in] 0, 1 [)

Example 2. Let's take

p = \frac{1}{2}, q = 1, r = \frac{3}{2}

.

L

equation then gives

Θ = \frac{9}{25}

and formula (32) shows us that the remainder of the quadrature formula

\int_{0}^{1} f (x) d x = \frac{1}{36} [2 f (0) + 25 f (\frac{9}{25}) + 9 f (1)] + R (f)

is given by the formula

\begin{aligned} R (f) = & \frac{\frac{1}{s + 1} - \frac{1}{36} [25 {(\frac{3}{5})}^{2 s} + 9]}{s (2 s - 1) (s - 1) (2 s - 3)} ξ^{2 - s} [3 f^{''} (ξ) + \\ + 12 ξ f^{'''} (ξ) + 4 ξ^{2} f^{' V} (ξ)] \end{aligned}

Or

ξ \in] 0, 1 [

And

s

being a number

> \frac{3}{2}

By taking

s = 21 e

stay

s

writing

R (f) = - \frac{1}{300} [3 f^{''} (ξ) + 12 ξ f^{'''} (Θ) + 4 ξ^{2} f^{' V} (ξ)]

BIBLIOGRAPHY
[1] S. Goła b, C. O lech: Contribution to the theory of the Simpsonian formula of approximate quadratures. Ann. Polon. Math. (1954) 176-183.
[2] G. P ol y a, G. S zeg ö: Aufgaben und Lehrsätze aus der Analysis, II. Berlin 1925.
[3] T. Popoviciu: Asupra unui determinant. Gas. Mast. Ser. A. 36 (1931) 405-408 (in Romanian).
[4] 'T. Popoviciu: Notes on higher order functions I. Mathematica (Cluj) 12 (1936) 81-92.
[5] T. Po poviciu: On the remainder in certain linear formulas of approximation of the analysis. Mathematica (Cluj) 24 (1959) 95-142。

CALCULATION INSTITUTE, Str. Republicii nr. 37, CLUJ, ROMANIA

- When at least two of the points $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ are equal to 0, the difference divided (29, (at least for $p ⩽ 1$ , is not defined.

1974

On a quadrature formula of S. Golab and C. Olech

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