On some minimizing polynomials

Tiberiu Popoviciu
(original), Approximation Theory, orthogonal polynomials, paper, polynomials

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T. Popoviciu

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T. Popoviciu, Sur certaines polynômes minimisants, Bull. de la Sect. Sci. de L’Acad. Roumaine, 12 (1929) no. 6, pp. 133-136 (in French).

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Bulletin de la Section Scientifique de l’Académie Roumaine

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1929 a -Popoviciu- Bull. Sect. Sci. - On certain minimizing polynomials
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ROMANIAN ACADEMY

NEWSLETTER

Twelfth Year
Section of the Scientific Section,
Twelfth Year
1929,
No. 6

ON CERTAIN MINIMIZING POLYNOMIES

BY

TIBERIU POPOVICIU

(Note presented to the Romanian Academy by Mr. G. Țițeica, MAR
I. Definition of the function) F ( z 1 , z 2 , … z n ) F z 1 , z 2 , … z n F(z_(1),z_(2),dotsz_(n))F\left(z_{1}, z_{2}, \ldots z_{n}\right)F(z1,z2,…zn)The real function F ( z 1 , z 2 , … z n ) F z 1 , z 2 , … z n F(z_(1),z_(2),dotsz_(n))F\left(z_{1}, z_{2}, \ldots z_{n}\right)F(z1,z2,…zn)of n n nnnreal variables z 1 , z 2 , … z n z 1 , z 2 , … z n z_(1),z_(2),dotsz_(n)z_{1}, z_{2}, ... z_{n}z1,z2,…znis determined as follows:
I. It is defined in the domain
( D ) z 1 ⩾ , o , z 2 ⩾ , 0 , … z n ⩾ 0 ( D ) z 1 ⩾ , o , z 2 ⩾ , 0 , … z n ⩾ 0 (D)quadz_(1) >= ,o,z_(2) >= ,0,dotsz_(n) >= 0(D) \quad z_{1} \geqslant, o, z_{2} \geqslant, 0, \ldots z_{n} \geqslant 0(D)z1⩾,o,z2⩾,0,…zn⩾0
and is positive in ( D ) ( D ) (D)(D)(D)2.
It is zero at the origin:
F ( 0 , … 0 ) = 0 F ( 0 , … 0 ) = 0 F(0,dots0)=0F(0, \ldots 0)=0F(0,…0)=0
This assumption is by no means essential, but it is convenient and in no way restricts the generality of the problem.
3. It is convex in ( D ) ( D ) (D)(D)(D)4.
It is increasing in ( D ) 1 ( D ) 1 (D)^(1)(D)^{1}(D)1).
It follows that the function is continuous in ( D ) ( D ) (D)(D)(D)and that if a variable z i z i z_(i)z_{i}zitends towards infinity F F FFFalso tends towards infinity.
2. Definition and uniqueness of the minimizing polynomial. We now consider n n nnndistinct points on the real axis:
x < x 2 < … < x n x < x 2 < … < x n x < x_(2) < dots < x_(n)x < x_{2} < \ldots < x_{n}x<x2<…<xn
and all polynomials of degree k < n k < n k < nk<nk<nwhose coefficient of the highest power of x x xxxis equal to I I III:
P ( x ) = x k + … P ( x ) = x k + … P(x) = x_(k) + dotsP(x) = x_{k} + ...P(x)=xk+…
the function is non-concave, respectively non-decreasing.
Let 1' be the expression:
F ( P ( x ) ) = F ( | P ( x 1 ) | , | P ( x 2 ) | , … | P ( x n ) | ) F ( P ( x ) ) = F P x 1 , P x 2 , … P x n F(P(x))=F(|P(x_(1))|,|P(x_(2))|,dots|P(x_(n))|)F(P(x))=F\left(\left|P\left(x_{1}\right)\right|,\left|P\left(x_{2}\right)\right|, \ldots\left|P\left(x_{n}\right)\right|\right)F(P(x))=F(|P(x1)|,|P(x2)|,…|P(xn)|)
We then have the following property.
There is one and only one polynomial of degree k < n k < n k < nk<nk<nfor which the expression F ( P ( x ) ) F ( P ( x ) ) F(P(x))F(P(x))F(P(x))reaches its minimum. This is the polynomial P k ( x ) P k ( x ) P_(k)(x)P_{k}(x)Pk(x).
The minimum found is zero if k = n k = n k=nk=nk=nbut not zero and therefore positive if k < n k < n k < nk<nk<n.
The roots of the equation:
(I) P k ( x ) = 0 (I) P k ( x ) = 0 {:(I)P_(k)(x)=0:}\begin{equation*} P_{k}(x)=0 \tag{I} \end{equation*}(I)Pk(x)=0
possess interesting properties. We find ϵ n ϵ n epsilonε<sub>n</sub>ϵneffect that:
I. All the roots of equation (I) are real and contained within the closed interval ( x 1 , x n x 1 , x n x_(1),x_(n)x_{1}, x_{n}x1,xn2.
In an open interval ( α , β α , β alpha, beta\alpha, \betaα,β) inside ( x 1 , x n x 1 , x n x_(1),x_(n)x_{1}, x_{n}x1,xn) and which contains i i iiipoints x i x i x_(i)x_{i}xithere is at most i + I i + I i+Ii+Ii+Iroots.
3. Any root that does not coincide with a point x i x i x_(i)x_{i}xiis simple.
4. The points x 1 , x n x 1 , x n x_(1),x_(n)x_{1}, x_{n}x1,xnpoints may possibly be roots, but only simple ones. x 2 , x 2 , … x n − ′ 1 x 2 , x 2 , … x n − ′ 1 x_(2),x_(2),dotsx_(n-)^(')_(1)x_{2}, x_{2}, \ldots x_{n-}{ }^{\prime}{ }_{1}x2,x2,…xn−′1can be multiple roots but the order of multiplicity cannot exceed 2.
If the function F F FFFadmits a continuous partial derivative with respect to z i z i z_(i)z_{i}ziand which cancels itself out for z = 0 z = 0 z=0z=0z=0the point x i x i x_(i)x_{i}xicannot be a root if i = I , n i = I , n i=I,ni=I, ni=I,n, and can only be a simple root if i = 2 , 3 , … n − I i = 2 , 3 , … n − I i=2,3,dots nIi=2,3, \ldots nIi=2,3,…n−I3.
An important special case. Suppose that the function F F FFFadmits continuous second partial derivatives throughout the domain ( D D DDDIt follows that the first-order partial derivatives are all positive or zero, but they cannot all be zero at the origin. For the second derivatives, it follows that the quadratic form:
∑ i , j = 1 n ξ i ξ j δ F δ z i δ z i ∑ i , j = 1 n   ξ i ξ j δ F δ z i δ z i sum_(i,j=1)^(n)xi_(i)xi_(j)(delta F)/(deltaz_(i)deltaz_(i))\sum_{i, j=1}^{n} \xi_{i} \xi_{j} \frac{\delta F}{\delta z_{i} \delta z_{i}}∑i,j=1nξiξjδFδziδzi
is defined and positive. We now prove the following property: The minimizing polynomial P k ( x ) P k ( x ) P_(k)(x)P_{k}(x)Pk(x)satisfies the equations.
(2) ∑ i = 1 n x i s | P k ( x i ) | P k ( x i ) ⋅ δ F δ z i = 0 s = 0 , I , 2 , … k − I ∑ i = 1 n   x i s P k x i P k ( x i ) ⋅ δ F δ z i = 0 s = 0 , I , 2 , … k − I quadsum_(i=1)^(n)x_(i)^(s)(|P_(k)(x_(i))|)/(P_(k)(xi))*(delta F)/(deltaz_(i))=0quad s=0,I,2,dots kI\quad \sum_{i=1}^{n} x_{i}^{s} \frac{\left|P_{k}\left(x_{i}\right)\right|}{P_{k}(xi)} \cdot \frac{\delta F}{\delta z_{i}}=0 \quad s=0, I, 2, \ldots kI∑i=1nxis|Pk(xi)|Pk(xi)⋅δFδzi=0s=0,I,2,…k−I
the partial derivatives being taken at the point
| P k ( x 1 ) | , | P k ( x 2 ) | , … | P k ( x n ) | . P k x 1 , P k x 2 , … P k x n . |P_(k)(x_(1))|,|P_(k)(x_(2))|,dots|P_(k)(x_(n))|.\left|P_{k}\left(x_{1}\right)\right|,\left|P_{k}\left(x_{2}\right)\right|, \ldots\left|P_{k}\left(x_{n}\right)\right| .|Pk(x1)|,|Pk(x2)|,…|Pk(xn)|.
It follows that Q k − 1 ( x ) Q k − 1 ( x ) Q_(k-1)(x)Q_{k-1}(x)Qk−1(x)being an arbitrary polynomial of degree k − I k − I k-Ik-Ik−Iwe have:
∑ i = I n Q k − 1 ( x i ) | P k ( x − i ) | P k ( x i ) δ F δ z i = 0 ∑ i = I n   Q k − 1 x i P k ( x − i ) P k x i δ F δ z i = 0 sum_(i=I)^(n)Q_(k-1)(x_(i))quad(|P_(k)(x-i)|)/(P_(k)(x_(i)))(delta F)/(deltaz_(i))=0\sum_{i=I}^{n} Q_{k-1}\left(x_{i}\right) \quad \frac{\left|P_{k}(x-i)\right|}{P_{k}\left(x_{i}\right)} \frac{\delta F}{\delta z_{i}}=0∑i=InQk−1(xi)|Pk(x−i)|Pk(xi)δFδzi=0
which is a kind of orthogonality property.
The polynomial P k ( x ) P k ( x ) P_(k)(x)P_{k}(x)Pk(x)is the only polynomial of degree k k kkksatisfying equations (2).
4. Properties analogous to mechanical quadrature. Let a polynomial G ( x ) G ( x ) G(x)\mathrm{G}(x)G(x)degree 2 k − I 2 k − I 2k-I2 k-I2k−Iat most. Let α 1 , α 2 , … α k α 1 , α 2 , … α k alpha_(1),alpha_(2),dotsalpha_(k)\alpha_{1}, \alpha_{2}, \ldots \alpha_{k}α1,α2,…αkthe roots of equation (I). We have the formula:
(3) ∑ i = 1 n G ( x i ) | P k ( x i ) | δ F δ z i = ∑ j = 1 k A j G ( α j ) (3) ∑ i = 1 n   G x i P k x i δ F δ z i = ∑ j = 1 k   A j G α j {:(3)sum_(i=1)^(n)(G(x_(i)))/(|P_(k)(x_(i))|)(delta F)/(deltaz_(i))=sum_(j=1)^(k)A_(j)G(alpha_(j)):}\begin{equation*} \sum_{i=1}^{n} \frac{G\left(x_{i}\right)}{\left|P_{k}\left(x_{i}\right)\right|} \frac{\delta F}{\delta z_{i}}=\sum_{j=1}^{k} A_{j} G\left(\alpha_{j}\right) \tag{3} \end{equation*}(3)∑i=1nG(xi)|Pk(xi)|δFδzi=∑j=1kHASjG(αj)
Or:
A i = ∑ i = 1 n 1 ( x i − a j ) P k ′ ( a i ) | P k ( x i ) | P k ( x i ) ⋅ δ F δ z i A i = ∑ i = 1 n   1 x i − a j P k ′ a i P k x i P k x i ⋅ δ F δ z i A_(i)=sum_(i=1)^(n)(1)/((x_(i)-a_(j)))P_(k)^(')(a_(i))(|P_(k)(x_(i))|)/(P_(k)(x_(i)))*(delta F)/(deltaz_(i))A_{i}=\sum_{i=1}^{n} \frac{1}{\left(x_{i}-a_{j}\right)} P_{k}^{\prime}\left(a_{i}\right) \frac{\left|P_{k}\left(x_{i}\right)\right|}{P_{k}\left(x_{i}\right)} \cdot \frac{\delta F}{\delta z_{i}}HASi=∑i=1n1(xi−hasj)Pk′(hasi)|Pk(xi)|Pk(xi)⋅δFδzi
This formula is true if:
I. The polynomial P k P k P_(k)P_{k}Pkdoes not cancel itself out at any point x i x i x_(i)x_{i}xi2.
In every respect x i x i x_(i)x_{i}xiOr P k P k P_(k)P_{k}Pkcancels out we have
(4) ∂ F ∂ z i = o pour z i = o (4) ∂ F ∂ z i = o  pour  z i = o {:(4)(del F)/(delz_(i))=o" pour "z_(i)=o:}\begin{equation*} \frac{\partial F}{\partial z_{i}}=o \text { pour } z_{i}=o \tag{4} \end{equation*}(4)∂F∂zi=o For zi=o
Indeed, in this case these points are excluded from formula (3).
3. At every point x i x i x_(i)x_{i}xiOr P k P k P_(k)P_{k}Pkcancels out, we have outside of (4).
lim z i → 0 I z i ⋅ δ F δ z i = A = bonné ( nécessairement ⩾ 0 ) lim z i → 0   I z i ⋅ δ F δ z i = A =  bonné  (  nécessairement  ⩾ 0 ) lim_(z_(i)rarr0)(I)/(z_(i))*(delta F)/(deltaz_(i))=A=" bonné "quad(" nécessairement " >= 0)\lim _{z_{i} \rightarrow 0} \frac{\mathrm{I}}{z_{i}} \cdot \frac{\delta F}{\delta z_{i}}=A=\text { bonné } \quad(\text { nécessairement } \geqslant 0)limitzi→0Izi⋅δFδzi=HAS= Good ( necessarily ⩾0)
In this case, the formula retains a precise meaning.
The numbers A j A j A_(j)A_{j}HASjare positive and independent of G ( x ) G ( x ) G(x)G(x)G(x); we can write:
A j = I [ ( P k ′ ( α j ) ] 2 ∑ i = I n | P k ( x i ) | ( x i − α j 2 ⋅ δ F δ z i A j = I P k ′ α j 2 ∑ i = I n   P k x i x i − α j 2 ⋅ δ F δ z i A_(j)=(I)/([(P_(k)^(')(alpha_(j))]^(2))sum_(i=I)^(n)(|P_(k)(x_(i))|)/((x_(i)-alpha_(j)^(2))*(delta F)/(deltaz_(i))A_{j}=\frac{I}{\left[\left(P_{k}^{\prime}\left(\alpha_{j}\right)\right]^{2}\right.} \sum_{i=I}^{n} \frac{\left|P_{k}\left(x_{i}\right)\right|}{\left(x_{i}-\alpha_{j}^{2}\right.} \cdot \frac{\delta F}{\delta z_{i}}HASj=I[(Pk′(αj)]2∑i=In|Pk(xi)|(xi−αj2⋅δFδzi
If the preceding conditions are not met, formula (3) is no longer true for G ( x ) G ( x ) G(x)\mathrm{G}(x)G(x)arbitrary, or else loses its simplicity, the derivatives of G ( x ) G ( x ) G(x)G(x)G(x)introducing themselves. ◻ ◻ ◻\square◻
5. Tchebyscheff-Jackson polynomials on n n nnnpoints. It is easily shown that there is one and only one polynomial of degree k k kkk
Π k ( x ) = x k + … Π k ( x ) = x k + … Pi_(k)(x)=x^(k)+dots\Pi_{k}(x)=x^{k}+\ldotsPik(x)=xk+…
such as
Max ( λ 1 | Π k ( x 1 ) | , λ 2 | Π k ( x 2 ) | , … λ n | Π k ( x n ) | 1 ) Max λ 1 Π k x 1 , λ 2 Π k x 2 , … λ n Π k x n 1 Max(lambda_(1)|Pi_(k)(x_(1))|,lambda_(2)|Pi_(k)(x_(2))|,dotslambda_(n)|Pi_(k)(x_(n))|^(1))\operatorname{Max}\left(\lambda_{1}\left|\Pi_{k}\left(x_{1}\right)\right|, \lambda_{2}\left|\Pi_{k}\left(x_{2}\right)\right|, \ldots \lambda_{n}\left|\Pi_{k}\left(x_{n}\right)\right|{ }^{1}\right)Max(λ1|Pik(x1)|,λ2|Pik(x2)|,…λn|Pik(xn)|1)or as small as possible.
Let's take:
F ( P ( x , m ) ; m ) = ∑ i = 1 n λ i m | P ( x i ) | m n | m > 1 , λ i < 0 F ( P ( x , m ) ; m ) = ∑ i = 1 n   λ i m P x i m n m > 1 , λ i < 0 {:F(P(x,m);m)=sum_(i=1)^(n)(lambda_(i)^(m)|P(x_(i))|m)/(n)quad|m > 1,lambda_(i) < 0\left.F(P(x, m) ; m)=\sum_{i=1}^{n} \frac{\lambda_{i}^{m}\left|P\left(x_{i}\right)\right| m}{n} \quad \right\rvert\, m>1, \lambda_{i}<0F(P(x,m);m)=∑i=1nλim|P(xi)|mn|m>1,λi<0
Either P k , m P k , m P_(k,m)P_{k, m}Pk,mthe minimizing polynomial of this expression.
It is easily shown that:
The polynomial P k , m P k , m P_(k),mP_{k}, mPk,mtends uniformly towards Π k Π k Pi_(k)\Pi_{k}PikFor m → ∞ m → ∞ m rarr oom \rightarrow \inftym→∞
It also follows that:
lim F ( P k , m ; m ) m = ϱ k lim F P k , m ; m m = ϱ k limroot(m)(F(P_(k),m;m))=ϱ_(k)\lim \sqrt[m]{F\left(P_{k}, m ; m\right)}=\varrho_{k}limitF(Pk,m;m)m=ϱk
Or: ϱ k = Max ( λ 1 , | Π k ( x 1 ) , λ 2 | Π k ( x 2 ) | … λ n | Π k ( x n ) ∣ ) ϱ k = Max λ 1 , Π k x 1 , λ 2 Π k x 2 … λ n Π k x n ∣ quadϱ_(k)=Max(lambda_(1),|Pi_(k)(x_(1)),lambda_(2)|Pi_(k)(x_(2))|dotslambda_(n)|Pi_(k)(x_(n))∣)\quad \varrho_{k}=\operatorname{Max}\left(\lambda_{1},\left|\Pi_{k}\left(x_{1}\right), \lambda_{2}\right| \Pi_{k}\left(x_{2}\right)\left|\ldots \lambda_{n}\right| \Pi_{k}\left(x_{n}\right) \mid\right)ϱk=Max(λ1,|Pik(x1),λ2|Pik(x2)|…λn|Pik(xn)|)
Paris, June 20, 1929.
  1. Max ( a 2 , a 2 , … a n a 2 , a 2 , … a n a^(2),a^(2),dotsa_(n)a^{2}, a^{2}, \ldots a_{n}has2,has2,…hasn) means the largest quantity ai.
  1. 1 1 ^(1){ }^{1}1That is to say that ( z ′ 1 , … , z ′ n ) , ( z ′ ′ 1 , z ′ ′ 2 , … z ′ ′ n ) z ′ 1 , … , z ′ n , z ′ ′ 1 , z ′ ′ 2 , … z ′ ′ n (z^(')_(1),dots,z^(')_(n)),(z^('')_(1),z^('')_(2),dotsz^('')_(n))\left(z^{\prime}{ }_{1}, \ldots, z^{\prime}{ }_{n}\right),\left(z^{\prime \prime}{ }_{1}, z^{\prime \prime}{ }_{2}, \ldots z^{\prime \prime}{ }_{n}\right)(z′1,…,z′n),(z′′1,z′′2,…z′′n)being two points of the domain ( D ) ( D ) (D)(D)(D)We have:
    F ( z 1 ′ + z 1 ′ ′ 2 , z 2 ′ + z 2 ′ ′ 2 , … z n ′ + z n ′ ′ 2 ) ⩽ 1 2 ( F ( z 1 ′ , z 2 ′ , … z n ′ ) + F ( z 1 ′ ′ , z 2 ′ ′ , … z n ′ ′ ) ) et F ( z 1 ′ , z 2 ′ , … z n ′ ) ⩽ F ( z 1 ′ ′ , z 2 ′ ′ , … z n ′ ′ ) si z i ′ ⩽ z i ′ ′ i = 1 , 2 , … n F z 1 ′ + z 1 ′ ′ 2 , z 2 ′ + z 2 ′ ′ 2 , … z n ′ + z n ′ ′ 2 ⩽ 1 2 F z 1 ′ , z 2 ′ , … z n ′ + F z 1 ′ ′ , z 2 ′ ′ , … z n ′ ′  et  F z 1 ′ , z 2 ′ , … z n ′ ⩽ F z 1 ′ ′ , z 2 ′ ′ , … z n ′ ′  si  z i ′ ⩽ z i ′ ′ i = 1 , 2 , … n {:[F((z_(1)^(')+z_(1)^(''))/(2),(z_(2)^(')+z_(2)^(''))/(2),dots(z_(n)^(')+z_(n)^(''))/(2)) <= (1)/(2)(F(z_(1)^('),z_(2)^('),dotsz_(n)^('))+F(z_(1)^(''),z_(2)^(''),dotsz_(n)^('')))],[" et "quad F(z_(1)^('),z_(2)^('),dotsz_(n)^(')) <= F(z_(1)^(''),z_(2)^(''),dotsz_(n)^(''))" si "z_(i)^(') <= z_(i)^('')quad i=1","2","dots n]:}\begin{gathered} F\left(\frac{z_{1}^{\prime}+z_{1}^{\prime \prime}}{2}, \frac{z_{2}^{\prime}+z_{2}^{\prime \prime}}{2}, \ldots \frac{z_{n}^{\prime}+z_{n}^{\prime \prime}}{2}\right) \leqslant \frac{1}{2}\left(F\left(z_{1}^{\prime}, z_{2}^{\prime}, \ldots z_{n}^{\prime}\right)+F\left(z_{1}^{\prime \prime}, z_{2}^{\prime \prime}, \ldots z_{n}^{\prime \prime}\right)\right) \\ \text { et } \quad F\left(z_{1}^{\prime}, z_{2}^{\prime}, \ldots z_{n}^{\prime}\right) \leqslant F\left(z_{1}^{\prime \prime}, z_{2}^{\prime \prime}, \ldots z_{n}^{\prime \prime}\right) \text { si } z_{i}^{\prime} \leqslant z_{i}^{\prime \prime} \quad i=1,2, \ldots n \end{gathered}F(z1′+z1′′2,z2′+z2′′2,…zn′+zn′′2)⩽12(F(z1′,z2′,…zn′)+F(z1′′,z2′′,…zn′′)) And F(z1′,z2′,…zn′)⩽F(z1′′,z2′′,…zn′′) if zi′⩽zi′′i=1,2,…n
    equality being possible only if z ′ i = z ′ ′ i i = 1 , 2 , … n z ′ i = z ′ ′ i i = 1 , 2 , … n z^(')_(i)=z^('')_(i)quad i=1,2,dots nz^{\prime}{ }_{i}=z^{\prime \prime}{ }_{i} \quad i=1,2, \ldots nz′i=z′′ii=1,2,…nOtherwise
1929

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