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T. Popoviciu, Sur la distribution des zeros de certains polynômes minimisants, Bul. de la Sect. Sci. de l’Acad, Roumaine, 16 (1934), pp. 214-217 (in French).

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1934 a -Popoviciu- Bull. Sect. Sci. Acad. Roum. – Sur la distribution des zeros de certains polynomes minimisants

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Bul. de la Sect. Sci. de l’Acad, Roumaine

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Romanian Society of Mathematical Sciences

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[Zbl 0010.30004, JFM 62.1222.03]

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1934 a -Popoviciu- Bull. Section. Sci. Acad. Rum. - On the distribution of zeros of certain polynomials

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ON THE DISTRIBUTION OF ZEROS IN CERTAIN MINIMIZING POLYNOMIES

TIBERIU POPOVICIU

Former student of the Ecole Normale Supérieure.

Note presented to the Romanian Academy by Mr. G. Tiţeica, MAR
I.

f (x)

a real, uniform function defined for the values

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

of the real variable

x

Let's consider the expression

\begin{matrix} (I) & E (f) = \sum_{i = 1}^{n} λ_{i} {(f (x_{i}))}^{2} \end{matrix}

λ_{1}, λ_{2}, \dots, λ_{n}

being positive coefficients. Among all the polynomials

P (x)

of degree

m < n

of the shape

x^{m} + \dots

there is one for which

E (P)

is the minimum. That is

P_{m}

this polynomial, which we can call the minimizing polynomial of degree

m

of expression (1). The polynomial

P_{m}

is such that for any other polynomial

Q (x)

of degree

< m

we have

\begin{matrix} (2) & \sum_{i = 1}^{n} λ_{i} P_{m} (x_{i}) Q (x_{i}) = 0 \end{matrix}

It is easy to deduce that

The polynomial $P_{m}$ has all its real zeros, distinct and located within the interval ( $x_{1}, x_{n}$ ).
The sequel
(3)

P_{m} (x_{1}), P_{m} (x_{2}), \dots P_{m} (x_{u})

presents at least

m

Sign variations.
In the following (3) we remove the zero terms. We will also say, to simplify the language, that the zeros of the polynomial

P_{m} (x)

separate the zeros of the polynomial

P_{n} (x) = (x - x_{1}) (x - x_{2}) \dots (x - x_{n})

Polynomials

P_{m} (x), P_{n} (x)

have at most

n - m - 1

common zeros.
2. Now let 1' be the expression

\begin{matrix} (4) & I (f) = \int_{a}^{b} p (x) (f (x))^{2} d x \end{matrix}

(

a, b

) being a finite interval and

p (x)

a non-negative summable function in

(a, b)

Let us now designate by

P_{n} (x)

the polynomial that minimizes the expression

I (P)

in the field of polynomials

P

of degree

n

of the shape

x^{n} + \dots

Let

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

the zeros of

P_{n}

which, as we know, are all real and distinct.

An arbitrary polynomial

Q (x)

of degree

< n

is written in the form

Q (x) = P_{n} (x) \sum_{i = 1}^{n} (\frac{Q (x_{i})}{(x - x_{i}) P_{n}^{'} (x_{i})})

Taking into account the orthogonality condition, we have
with

I (Q) = E (Q)

λ_{i} = \int_{a}^{b} p (x) {(\frac{P_{n} (x)}{(x - x_{i}) P_{n}^{'} (x_{i})})}^{2} \cdot d x > 0, i = 1, 2, \dots, n

We deduce the following property:
The zeros of a polynomial in the sequence of minimizing polynomials (orthogonal polynomials)

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

are separated by the zeros of any preceding polynomial.
Note: The property remains true if we substitute into expression (4)

d x

d α (x), α (x)

being a non-decreasing function and if we take Stieltjes integrals (expression (I) is of this form). We can also consider an infinite interval. We choose, of course, the functions

α (x), p (x)

so that the integrals involved in determining the minimizing polynomials are meaningful (that the moments exist).
3. Let

F (x)

a polynomial of degree

n

having all its zeros real and distinct, and let us form the Sturm sequence of this polynomial

\begin{matrix} (5) & F (x), F_{1} (x), F_{2} (x), \dots, F_{n} (x) (F_{1} (x) = F^{'} (x)) \end{matrix}

KRONECKER demonstrated

^{1}

) that if

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

are the zeros of

F (x)

we have

\sum_{i = 1}^{n} \frac{F_{k} (x_{i}) F_{h} (x_{i})}{{(F^{'} (x_{i}))}^{2}} = 0 (k \neq h)

which shows that sequence (5) is orthogonal, therefore.

If the zeros of the polynomial

F (x)

are all real and distinct, the zeros of a polynomial in the STURM sequence (5) are separated by the zeros of any polynomial that follows it.

Note. The property remains true if

F_{1} (x)

is a polynomial of degree

n

-I whose zeros separate those of

F (x)

4.
Let us consider more generally a sequence of functions

f_{0} (x), f_{1} (x), \dots f_{n} (x), \dots

uniform and continuous in the interval (

a, b

) and suppose that for all

n

and for any sequence of points

ξ_{0} < ξ, < ξ_{2} \dots < ξ_{n}

, of (

a, b

) the determinant

\begin{matrix} | f_{0} (ξ_{i}) f_{1} (ξ_{i}) \dots f_{n} (ξ_{i}) | \\ i = 0, 1, 2, \dots, n \end{matrix}

either

> O

We then know that a linear combination of the form

\begin{aligned} (6) & φ_{n} = c_{0} f_{0} + c_{1} f_{1} + \dots + c_{n} f_{n} \\ (c_{0}, c_{1}, \dots, c_{n} constantes) \end{aligned}

enjoys the following properties
I.

φ_{n}

is completely determined by its values in

n + 1

points of the interval (

a, b

2 .

φ_{n}

cancels out at most

n

times in (

a, b

3.
if

φ_{n}

cancels out

n

times in (

a, b

) it changes sign at each of its zeros.

Let's now express

J (f) = \int_{a}^{b} p (x) | f (x) |^{q} d x, q > I

p (x)

being the function defined above.
Among all linear combinations of the form (6), with

c_{n} = I

, there is one for which

J (φ_{n})

is the minimum. That is

φ_{n}^{(q)}

this linear combination.

We have

^{1}

)

\int_{b}^{a} p {| φ_{n}^{(q)} |}^{q - 1} (s g \cdot φ_{n}^{(q)}) φ_{n - 1} d x = 0

φ_{n - 1}

is any linear combination between the

n

first functions

f_{i}

It follows that

φ_{n}^{(q)}

change sign at least

n

three in (

a, b

). We also see that the expression

α {| φ_{n}^{(q)} |}^{q - I} sg \cdot φ_{n}^{(q)} + β {| φ_{n}^{(q)} r |}^{q - I} sg \cdot φ_{n - I}^{(q)}

change sign at least

n - 1

times in (

a, b

) whatever

α

And

β

From this
it follows that

α φ_{n}^{(q)} + β φ_{n - 1}^{(q)}

change sign at least

n

-I times in (

a, b

) whatever a and

β

We can therefore conclude, as OD KELLOGG does for the case

q = 2^{1}

), that

The zeros of

φ_{n}^{(q)}

are separated by those of

φ_{n}^{(q)}^{-}

5.
Let us consider again the expression

J (f - φ_{n}) f (x)

being a continuous function. Among all linear combinations of the form (6) there exists one - which we will denote by

Φ_{n}^{(q)}

- for which

J

(

f - φ_{n}

) reaches its minimum.

As above, we show that the difference

{| f - Φ_{n}^{(q)} |}^{q - r} (sg . (f - Φ_{u}^{(q)})) - {| f - Φ_{n - I}^{q} |}^{q - I} (sg . (f - Φ_{n - I}^{(q)}))

and consequently also the difference

Φ_{n}^{(q)} - Φ_{n - r}^{(q)}

change sign at least

n

-I times in (

a, b

Let's return to the simple case of polynomials
, when the sequence of functions

f_{i}

reduces to

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

We then see that the equation

Φ_{n}^{(q)} - Φ_{n - I}^{(q)} = 0

has all its real roots.
We know that if

q \to \infty, Φ_{n}^{(q)}

tends uniformly towards the Chebyshef polynomial of degree

n

of the function

f (x)

, SO

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

are the Chebyshef polynomials of best approximation of the continuous function

f (x)

in the meantime

(a, b), l^{'} e ́ -

equation of degree

n

T_{n} - T_{n - 1} = 0

has all its real roots.

$^{1)}$ See G. MIGNOSI “Teorema di Sturm e sue estensioni” Rendiconti Circ. Mast. Palermo t. XLIX (1925) p. 84.
$^{1)}$ See the works of MD Jackson. Especially his book "The Theory of Approximation" New York 1930.
$^{1}$ ) OD KELLOGG "The Oscillation of Function of an Orthogonal Set”. Amer. Jurn. of Math. XXXVIII (1916) p. 1.

Posts by Tiberiu Popoviciu

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About this paper

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Paper (preprint) in HTML form

ON THE DISTRIBUTION OF ZEROS IN CERTAIN MINIMIZING POLYNOMIES

TIBERIU POPOVICIU

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