Remark on the polynomials of best approximation

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Tiberiu Popoviciu

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T. Popoviciu

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T. Popoviciu, Remarque sur les polynômes de meilleure approximation, Buletinul Soc. de Ştiinţe din Cluj, 5 (1930), pp.279-286 (in French).

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Buletinul Soc. de Ştiinţe din Cluj

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[JFM 57.1428.01]

Republished in: T. Popoviciu, Remarque sur les polynômes de meilleure approximation, Mathematica, 4 (1930), pp. 73-80 (in French)

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1930 c -Popoviciu- Bull. Soc. Sc. Cluj - Remark on best approximation polynomials

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BULETINUL SOCIETĂTII DE ŞTINTE DIN CLUJ (ROMÂNIA) BULLETIN OF THE SOCIETY OF SCIENCES OF CLUJ (ROMANIA) Volume V,

1^{D}

Part, pp. 279-286. June 5, 1930.

NOTE ON BEST APPROXIMATION POLYNOMIALS

1 per

Tiberiu Popoviciu

Student at the École Normale Supérieure, Paris.
Admitted on February 21, 1930.

We consider the class (b) of functions defined and continuous in the finite and closed interval $(has, b); (has < b)$ . Regarding the properties of these functions, let us remember that they can vanish at a finite or infinite number of points, but the set of these points always forms a finite number of intervals which can also be reduced to a single point.

Let us now consider a system of

n + 1

functions of class (b)
(1)

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

linearly independent, that is to say an equality such as
(2)

P = c_{0} f_{0} + c_{1} f_{1} + \dots + c_{n} f_{n} = 0

c_{0}, c_{1}, \dots, c_{n}

being constants, cannot take place everywhere in

(has, b)

that if

c_{0} = c_{1} = \dots = c_{n} = 0

Note that system (1) may not be linearly independent in an interval

(α, β)

interior to

(has, b)

[such as

| has - α | + | β - b | \neq 0

].

Any homogeneous linear expression such as (2) will be called a polynomial of the system (1). To any polynomial

P

we can make a point correspond

M

coordinates

c_{0}, c_{1}, \dots, c_{n}

in ordinary space

n + 1

dimensions. We call the point M the image of the corresponding polynomial P.

The fact that the functions (1) are linearly independent can be expressed as follows:

He

y

a one-to-one correspondence between polynomials

P

and their images

M

.

For the points where

P

If this is cancelled, we make the following agreements:

(^{1})

10. We call a simple zero a point inside the interval (

has, b

) where P becomes zero by changing sign, or an endpoint (

has

or b) where P is zero.

2^{0}

We call a double zero, and we count it as two single zeros, an interior point where

P

cancels out without changing sign.

A system (1) is a Chebyshev system or system (T) if any polynomial

P

cannot have more

n

zeros. It follows that for a system not to be a system (T), it is necessary and sufficient that there be at least one polynomial having at least

n + 1

zeros.

For example, the system

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

is a system (T) in any finite interval. Conversely, the system

1, si x, \cos x, \dots, si n x, \cos n x

is not a system (T) in every interval (

0, 2 π

), but there is one in an interval of shorter length than

2 π

2.
We will demonstrate the following auxiliary property:

If (1) is not a system (

T

) we can find

n + 1

distinct points

x_{1}, x_{2}, \dots, x_{n_{- 1}}

completely inside (

has, b

) such as the determinant:

\begin{matrix} (3) & | \begin{array}{cccc} f_{0} (x_{1}) & f_{1} (x_{1}) & \dots & f_{n} (x_{1}) \\ f_{0} (x_{2}) & f_{1} (x_{2}) & \dots & f_{n} (x_{2}) \\ \cdot & \cdot & \cdot & \cdot \\ f_{0} (x_{n + 1}) & f_{1} (x_{n + 1}) & \dots & f_{n} (x_{n + 1}) \end{array} | \end{matrix}

is zero.
Indeed, there exists a polynomial having at least

n + 1

zeros. If such a polynomial vanishes at at least

n + 1

distinct points, none of which coincides with an endpoint, the property is obvious.

The points where P vanishes can be arranged into 3 categories.
10. Or P vanishes while changing sign; let

\begin{matrix} (4) & x_{1}, x_{2}, \dots, x_{m} \end{matrix}

these points.

2^{0}

Or P vanishes without changing sign; let
(5)

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

(1) S. Bernstein "Lectures on Extremal Properties ... etc." p. 1; J.W. Young "General Theory of Approximation by Functions Involving a Given Number of Arbitrary Parameters" Transactions of the Amer. Math. Soc. 8 (1907), p. 331.
These points. Finally:

3^{0}

The extremes where P vanishes

x^{''}_{1}, x^{''}_{2}, \dots, x^{''}_{i} (i = 0, 1, 2)

We have, by hypothesis:

m + 2 k + i ≧ n + 1

Suppose that:

m + k < n + 1

therefore, all the more so

k < n + 1

Let us first assume that
and consider the table:

k < n

\begin{matrix} (7) & ‖ \begin{array}{cccc} f_{0} (x^{'}_{1}) & f_{1} (x^{'}_{1}) & \dots & f_{n} (x^{'}_{1}) \\ f_{0} (x^{'}_{2}) & f_{1} (x^{'}_{2}) & \dots & f_{n} (x^{'}_{2}) \\ ⋮ \\ f_{0} (x^{'}_{k}) & f_{1} (x^{'}_{k}) & \dots & f_{n} (x^{'}_{k}) \\ f_{0} (x^{''}_{1}) & f_{1} (x^{''}_{1}) & \dots & f_{n} (x^{''}_{1}) \\ f_{0} (x^{''}_{i}) & f_{1} (x^{''}_{i}) & \dots & f_{n} (x^{''}_{i}) \end{array} ‖ \end{matrix}

which contains at least as many columns as rows.
If in table (7) all the determinants formed by

k + i

If any columns are zero, the desired property follows. But suppose there is a non-zero determinant, for example:

\begin{matrix} (8) & \begin{array}{cccc} f_{0} (x_{1}^{'}) & f_{1} (x_{1}^{'}) & \dots & f_{k + i} (x_{1}^{'}) \\ f_{0} (x_{2}^{'}) & f_{1} (x_{2}^{'}) & \dots & f_{k + i} (x_{2}^{'}) \\ \cdot & \cdot & \cdot & \cdot \\ f_{0} (x_{i}^{''}) & f_{1} (x_{i}^{''}) & \dots & f_{k + i} (x_{i}^{''}) \end{array} | \neq 0 \end{matrix}

At each point

x^{'};

And

x^{''};

let's assign a non-zero number

a^{'};

resp.

a^{''}_{j}

such that its sign is that of the polynomial P in the neighborhood of

x^{'}_{j}

resp.

x^{''}

; [the neighborhood of an endpoint is counted only towards the interior of the interval]. We can then determine a system of non-zero numbers
such that

λ_{0}, λ_{1}, \dots, λ_{k + 1}

\begin{aligned} λ_{0} f_{0} (x_{1}^{'}) + λ_{1} f_{1} (x_{1}^{'}) + \dots + λ_{k + i} f_{k + i} (x_{1}^{'}) = a_{1}^{'} \\ λ_{0} f_{0} (x_{2}^{'}) + λ_{1} f_{1} (x_{2}^{'}) + \dots + λ_{k + i} f_{k + i} (x_{2}^{'}) = a_{2}^{'} \\ \dots \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \end{aligned}

λ_{0} f_{0} (x_{i}^{''}) + λ_{1} f_{1} (x^{''}) + \dots + λ_{k + i} f_{k + i} (x^{''}_{i}) = a_{i}^{''}

Either

ε > 0

and consider the polynomial

P_{1} = P - ε (λ_{0} f_{0} + λ_{1} f_{1} + \dots + λ_{k + i} f_{k + i})

If

ε = 0

We have

P_{1} \equiv P

. If

ε

is small but not zero.

P_{1}

is close to P. We are sure that if we take

ε

small enough, at each zero

x_{j}

P will correspond to a zero

x_{j}

of

P_{1}

neighbor of

x_{j}

We can also see this because of the choice of numbers.

a^{'}_{j}, a^{''}_{j}

that at each zero

x^{'}_{j}

correspond to at least two simple zeros for

P_{1}

and to zero

x^{''}_{j}

corresponds to at least one zero inside (

a, b

).

So, finally,

ε

being sufficiently small, the polynomial

P_{1}

cancels out at least

m + 2 k + i

times; the property results immediately. In the case

k = n

it suffices to introduce at most one point

x_{l}^{''}

sion
:
3. Let

f (x)

a function of class (b) and consider the express-

I (c_{0}, c_{1}, \dots, c_{n}) = {Max}_{(a, b)} | f - c_{0} f_{0} - c_{1} f_{1} - \dots - c_{n} f_{n} |

To simplify the writing we can write

I (c_{0}, c_{1}, \dots, c_{n}) = {Max}_{(a, b)} | f - P | = I (M)

Let M be the image of the polynomial P. Then, the function I(M) is continuous with respect to the coefficients.

c_{0}, c_{1}, \dots, c_{n}

for everything

a \leq x \leq b

.

The expression I(M) has a lower limit, and we know that this limit is attained by at least one polynomial P, therefore by at least one finite value system.

γ_{0}, γ_{1}, \dots, γ_{n}

quantities

c_{0}, c_{1}, \dots c_{n}

:

lim I (M) = I (M_{1}) = I (γ_{0}, γ_{1}, \dots, γ_{n})

M_{1}

is then a minimizing point and the corresponding polynomial is a minimizing polynomial. In general, it is difficult to decide whether the minimizing polynomial is unique or not, but:

If there are two distinct minimizing polynomials, there are infinitely many.

Indeed

M_{1}, M_{2}

two distinct minimizing points and consider the point

M_{3} = \frac{M_{1} + λ M_{2}}{1 + λ}, λ \geq 0

on the segment

M_{1} M_{2}

The symbolic representation above being clear.

It is easy to see that:

I (M_{3}) \leq \frac{I (M_{1}) + λ I (M_{2})}{1 + λ} = I (M_{1})

but by hypothesis,
therefore

\begin{aligned} I (M_{3}) \geq I (M_{1}) \\ I (M_{3}) = I (M_{1}) \end{aligned}

It follows that any point of the segment

M_{1} M_{2}

is minimizing. On the other hand, every minimizing point is at a finite distance, because for a point with unbounded coordinates I(M) becomes infinite and at the same time:

lim I (M) \leq {Max}_{(a, b)} | f | = quantité bornée.

We can therefore state the property:
Minimizing points form a closed and bounded convex domain.
4. We propose to prove the following property:

The necessary and sufficient condition for there to be only one minimizing polynomial, regardless of the function

f

of class (b), is that the system (1) is a system (T).

We know that this condition is sufficient (

^{1}

We will show that if the system is not a Chebyshev system, we can find a function

f

admitting an infinite number of minimizing polynomials.

We know that there is a polynomial

P = c_{0} f_{0} + c_{1} f_{1} + \dots + c_{n} f_{n}

canceling out in at least

n + 1

distinct points within the interval (

a, b

). If

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

are

n + 1

points satisfying this condition, the determinant (3) is zero because we assume, of course, that

P \equiv\equiv 0

.

We can then find

n + 1

Numbers

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

such as the system

\begin{array}{ll} E_{1} = \overset{―}{c_{0}} f_{0} (x_{1}) + \overset{―}{c_{1}} f_{1} (x_{1}) + \dots & + \overset{―}{c_{n}} f_{n} (x_{1}) - a_{1} = 0 \\ E_{2} = \overset{―}{c_{0}} f_{0} (x_{2}) + \overset{―}{c_{1}} f_{1} (x_{2}) + \dots & + \overset{―}{c_{n}} f_{n} (x_{2}) - a_{2} = 0 \\ E_{n + 1} = \overset{―}{c_{0}} f_{0} (x_{n + 1}) + \overset{―}{c_{1}} f_{1} (n + 1) + \dots + \overset{―}{c_{n}} f_{n} (x_{n + 1}) - a_{n + 1} = 0 \end{array}

is incompatible. In this case, the expression

Max (| E_{1} |, | E_{2} |, \dots, | E_{n + 1} |) (^{2})

which is a continuous function of

{\bar{c}}_{0}, {\bar{c}}_{1}, \dots, {\bar{c}}_{n}

, admits a non-zero positive minimum. Let

m

the value of this minimum; it is attained for an infinite number of value systems

{\bar{c}}_{0}, {\bar{c}}_{1}, \dots, {\bar{c}}_{n}

because if

\overset{―}{c_{i}} = c_{i}^{*} i i = 0, 1, \dots, n

(') S. Bernstein loc. cit. p. 3; JW Joung loc. cit.
(

^{2}

)

Max (λ_{1}, λ_{2}, \dots, λ_{n}

) means the value of the largest quantity

λ i

.
is a minimizing system, the systems

c_{i} = c_{i}^{*} + λ c_{i} i = 0, 1, 2, . ., n

will also minimize

λ

being arbitrary.
Now if

f

is a continuous function taking the values

a_{i}

to the points

x_{i} (i = 1, 2, \dots, n + 1)

, we will have for every polynomial

Q

Max | f - Q | \geq m [dans (a, b)] .

Let's ask

P^{*} = c_{0}^{*} f_{0} + c_{1}^{*} f_{1} + \dots + c_{n}^{*} f_{n}

[one can also have

P^{*} \equiv 0

To clarify, let's assume that the polynomial P, taken initially, vanishes at a finite number of points; these are first the points

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

and then some other points

x^{'}_{1}, x^{'}_{2}, \dots, x^{'}_{r}

(which may not exist at all).

Let's take the function

f

such that:
(9)

\begin{matrix} (10) & \begin{array}{cc} f (x_{i}) = P^{*} (x_{i}) - a_{i} & i = 1, 2, \dots, n + 1 \\ f (x_{i}^{'}) = 0 & i = 1, 2, \dots, r \end{array} \end{matrix}

\begin{aligned} Les points x_{i}, x_{i}^{'} partagent l'intervalle (a, b) en un certain nombre \\ us-in-in \end{aligned}

of subintervals (

n + r, n + r + 1

Or

n + r + 2

) such that in each of them the polynomial P retains a constant sign.

Either

{Max}_{(a, b)} | P | = A

Let's take

0 > λ > - \frac{m}{A}

SO

| λ P | < m

We determine

f

in each subinterval such that it is continuous and such that (9) and (10) are satisfied. To do this, we will examine the various kinds of subintervals that can arise.
10. Interval

(a, x^{'}_{1})

Within such an interval, we will take

f = \frac{m}{2 A} P

then in (

a, x^{'}_{1}

)
And

λ < 0, | f - λ P | = | f | + | λ P | = {\frac{m}{2 A} + | λ |} | P |

| f - λ P | < m

provided that

- \frac{m}{2 A} < λ < 0

The same construction is used for an interval (

x^{'}_{r}, b

).

2^{0}

. Interval

(a, x_{1})

[Or

(x_{n + 1}, b)]

. We have

| P^{*} (x_{1}) - a_{1} | \leq m

We will take from (

a, x_{1}

)

\begin{array}{lll} f = P^{*} (x_{1}) - a_{1} - \frac{m}{A} P & si & P [P^{*} (x_{1}) - a_{1}] > 0 \\ f = P^{*} (x_{1}) - a_{1} & si & P [P^{*} (x_{1}) - a_{1}] < 0 \end{array}

we then see that
(11)

Max | f + λ P | \leq m

provided that
(12)

0 > λ > - \frac{m}{A}

Interval ( $x_{1}, x_{1}^{'}$ ).

We can take

\begin{array}{ll} f = (x - x_{1}^{'}) \frac{P^{*} (x_{1}) - a_{1}}{x_{1} - x_{1}^{'}} - \frac{m}{A} P & si P [P^{*} (x_{1}) - a_{1}] > 0 \\ f = (x - x_{1}^{'}) \frac{P^{*} (x_{1}) - a_{1}}{x_{1} - x_{1}^{'}}, & si P [P^{*} (x_{1}) - a_{1}] < 0 \end{array}

and we still have (11) under conditions (12).
4). Interval

(x^{'}_{1}, x^{'}_{2})

All you have to do is take

f = - \frac{m}{A} P

and (11) will be satisfied with (12).
50. Interval (

x_{1}, x_{2}

Let's assume

g = \frac{x [P^{*} (x_{1}) - a_{1} - P^{*} (x_{2}) + a_{2}] + x_{2} ⌊ P^{*} (x_{1}) - a_{1} ⌉ - x_{1} [P^{*} (x_{2}) - a_{2} ⌉}{x_{1} - x_{1}}

We then ask

\begin{array}{ll} f = g - \frac{m}{A} P & si P [P^{*} (x_{1}) - a_{1}] > 0, P [P^{*} (x_{2}) - a_{2}] > 0 \\ ou bien si [P^{*} (x_{1}) - a_{1}] [P^{*} (x_{2}) - a_{2}] < 0 \\ f = g & si P [P^{*} (x_{1}) - a_{1}] > 0, P [P^{*} (x_{2}) - a_{2}] < 0 . . \end{array}

We can still determine one

λ_{1} > 0

such that if

0 > λ > - λ_{1}

we still have
in the interval (

x_{1}, x_{2}

(）．

Max | f - λ P | \leq m

This property results from the following observation:
Let

g (x)

a continuous function in

(0, 1)

(we take the inter-nalle

(0, 1)

to simplify the explanation) positive (negative) in this interval and

g (0) = g (1) = 1

So, to clarify things,

a^{'} > 0, b^{'} < 0

We can then determine a

λ_{1} > 0

such as for

0 < λ < λ_{1}

we have

a^{'} + x (b^{'} - a^{'}) + λ g < a

In

(0, 1)

The demonstration is immediate.
The function

f

Constructed in this way, it answers the question, because

λ

being between

0, - λ_{1}

Or

λ_{1}

is a positive number, we have

| f - λ P | \leq m

In

(a, b)

and at least in one respect

x_{i}

on al＇égalité

| f - λ P | = m .

On the other hand, any other polynomial gives

| f - Q | \geq m .

5. If P were to vanish throughout an entire interval (

α, β

(For example, we could still do the construction very easily.) The points

x_{1}, x_{2}

，

\dots . x_{n + 1}

can be taken inside

(α, β)

So apart from

(α, β)

We are doing the same construction as before. In (

α, β

(we still keep the conditions (9); and in each interval

x_{i}, x_{i + 1}

we take the function

t

linear; and constant in

(α, x_{1})

And(

x_{n + 1}, β

The function thus obtained still answers the question.

1930

Remark on the polynomials of best approximation

Abstract

Authors

Keywords

Paper coordinates

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About this paper

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DOI

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NOTE ON BEST APPROXIMATION POLYNOMIALS

Tiberiu Popoviciu

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