On a maximum problem of Stieltjes

Tiberiu Popoviciu
Uncategorized

Abstract

Authors

T. Popoviciu

Keywords

?

Paper coordinates

 T. Popoviciu, Sur un problème de maximum de Stieltjes, (French) C. R. Acad. Sci., Paris 202, 1645-1647 (1936).

PDF

1936 c -Popoviciu- Comptes Rendus – Sur un probleme de maximum de Stieltjes

About this paper

Journal

Comptes rendus de l’Académie des Sciences

Publisher Name
DOI
Print ISSN
Online ISSN

Zbl 0014.20902

google scholar link

??

Paper (preprint) in HTML form

1936 c -Popoviciu- Reports - On a Stieltjes maximum problem
Original text
Rate this translation
Your feedback will be used to help improve Google Translate



Theory of functions. - On a problem of Sticlijes maximum. Note by Mr. Tibère Popoviciu.
  1. Assumptions made about the function f ( x ) . − f ( x ) f ( x ) . − f ( x ) f(x).-f(x)f(x) .-f(x)f(x).−f(x)is a real, defined and uniform function on the set E formed by m m mmmfinite and closed intervals ( has i , b i has i , b i a_(i),b_(i)a_{i}, b_{i}hasi,bi) , has i < b i ( i = 1 , … , m ) ; b i ≦ has i + 1 ( i = 1 , … , m − 1 ) has i < b i ( i = 1 , … , m ) ; b i ≦ has i + 1 ( i = 1 , … , m − 1 ) a_(i) < b_(i)(i=1,dots,m);b_(i) <= a_(i+1)(i=1,dots,m-1)a_{i}<b_{i}(i=1, \ldots, m); b_{i} \leqq a_{i+1}(i=1, \ldots, m-1)hasi<bi(i=1,…,m);bi≦hasi+1(i=1,…,m−1)of the real axis. We will say that f ( x ) f ( x ) f(x)f(x)f(x)is a function (C) if, in each interval ( has i , b i ) : I ∘ has i , b i : I ∘ (a_(i),b_(i)):I^(@)\left(a_{i}, b_{i}\right): \mathrm{I}^{\circ}(hasi,bi):I∘it is non-negative in the open interval, but there are points as close as one wants to has i has i a_(i)a_{i}hasiand of b i b i bi)bi}biwhere the function is positive; 2 ∘ 2 ∘ 2^(@)2^{\circ}2∘it is continuous in has i has i a_(i)a_{i}hasiand in b i ; 3 ∘ f ( has i ) = f ( b i ) = 0 ; 4 ∘ b i ; 3 ∘ f has i = f b i = 0 ; 4 ∘ b_(i);3^(@)f(a_(i))=f(b_(i))=0;4^(@)b_{i}; 3^{\circ} f\left(a_{i}\right)=f\left(b_{i}\right)=0 ; 4^{\circ}bi;3∘f(hasi)=f(bi)=0;4∘it is exponentially concave in the whole interval. This last property means
    that
f ( x 2 ) > f ( x 1 ) x 3 − x 2 x 3 − x 1 f ( x 3 ) x 2 − x 1 x 3 − x 1 ( has i ≦ x 1 < x 2 < x 3 ≦ b i ) f x 2 > f x 1 x 3 − x 2 x 3 − x 1 f x 3 x 2 − x 1 x 3 − x 1 has i ≦ x 1 < x 2 < x 3 ≦ b i f(x_(2)) > f(x_(1))^((x_(3)-x_(2))/(x_(3)-x_(1)))f(x_(3))^((x_(2)-x_(1))/(x_(3)-x_(1)))quad(a_(i) <= x_(1) < x_(2) < x_(3) <= b_(i))f\left(x_{2}\right)>f\left(x_{1}\right)^{\frac{x_{3}-x_{2}}{x_{3}-x_{1}}} f\left(x_{3}\right)^{\frac{x_{2}-x_{1}}{x_{3}-x_{1}}} \quad\left(a_{i} \leqq x_{1}<x_{2}<x_{3} \leqq b_{i}\right)f(x2)>f(x1)x3−x2x3−x1f(x3)x2−x1x3−x1(hasi≦x1<x2<x3≦bi)
so that log ⁡ f ( x ) log ⁡ f ( x ) log f(x)\log f(x)log⁡f(x)is concave.
2. Assumptions made on the points x i x i x_(i)x_{i}xi. - We will say that the n n nnnpoints x 1 x 1 x_(1)x_{1}x1, x 2 , … , x n x 2 , … , x n x_(2),dots,x_(n)x_{2}, \ldots, x_{n}x2,…,xnof the real axis have a given distribution [ k 1 , k 2 , … , k m ] if k 1 , k 2 , … , k m if [k_(1),k_(2),dots,k_(m)]if\left[k_{1}, k_{2}, \ldots, k_{m}\right] \mathrm{si}[k1,k2,…,km]if, among these points, there are k i k i k_(i)k_{i}kiin the meantime ( has i , b i ) ( i = 1 , … , m ) has i , b i ( i = 1 , … , m ) (a_(i),b_(i))(i=1,dots,m)\left(a_{i}, b_{i}\right)(i=1, \ldots, m)(hasi,bi)(i=1,…,m). We have k i ≧ 0 , k 1 + k 2 + … + k m = n k i ≧ 0 , k 1 + k 2 + … + k m = n k_(i) >= 0,k_(1)+k_(2)+dots+k_(m)=nk_{i} \geqq 0, k_{1}+k_{2}+\ldots+k_{m}=nki≧0,k1+k2+…+km=n. We will say that two different distributions [ k 1 , k 2 , … , k m ] k 1 , k 2 , … , k m [k_(1),k_(2),dots,k_(m)]\left[k_{1}, k_{2}, \ldots, k_{m}\right][k1,k2,…,km]And [ k 1 ′ , k 2 ′ , … , k m ′ ] k 1 ′ , k 2 ′ , … , k m ′ [k_(1)^('),k_(2)^('),dots,k_(m)^(')]\left[k_{1}^{\prime}, k_{2}^{\prime}, \ldots, k_{m}^{\prime}\right][k1′,k2′,…,km′]of n n nnnpoints x 1 < x 2 < … < x n x 1 < x 2 < … < x n x_(1) < x_(2) < dots < x_(n)x_{1}<x_{2}<\ldots<x_{n}x1<x2<…<xn(none of which coincide with a has i has i a_(i)a_{i}hasior a b i b i bi)bi}bi) And x 1 ′ , x 2 ′ , … , x n ′ x 1 ′ , x 2 ′ , … , x n ′ x_(1)^('),x_(2)^('),dots,x_(n)^(')x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{n}^{\prime}x1′,x2′,…,xn′respectively, are consecutive if we can choose the points x i ′ x i ′ x_(i)^(')x_{i}^{\prime}xi′so that we have
x 1 < x 1 ′ < x 2 < x 2 ′ < … < x n < x n ′ x 1 < x 1 ′ < x 2 < x 2 ′ < … < x n < x n ′ x_(1) < x_(1)^(') < x_(2) < x_(2)^(') < dots < x_(n) < x_(n)^(')quadx_{1}<x_{1}^{\prime}<x_{2}<x_{2}^{\prime}<\ldots<x_{n}<x_{n}^{\prime} \quadx1<x1′<x2<x2′<…<xn<xn′Or x 1 ′ < x 1 < x 2 ′ < x 2 < … < x n ′ < x n x 1 ′ < x 1 < x 2 ′ < x 2 < … < x n ′ < x n quadx_(1)^(') < x_(1) < x_(2)^(') < x_(2) < dots < x_(n)^(') < x_(n)\quad x_{1}^{\prime}<x_{1}<x_{2}^{\prime}<x_{2}<\ldots<x_{n}^{\prime}<x_{n}x1′<x1<x2′<x2<…<xn′<xn.
We will also say that two distributions [ L 1 , L 2 , … , L m ] L 1 , L 2 , … , L m [l_(1),l_(2),dots,l_(m)]\left[l_{1}, l_{2}, \ldots, l_{m}\right][L1,L2,…,Lm]And [ h 1 , h 2 , … , h m ] h 1 , h 2 , … , h m [h_(1),h_(2),dots,h_(m)]\left[h_{1}, h_{2}, \ldots, h_{m}\right][h1,h2,…,hm]of n n nnnpoints x 1 < x 2 < … < n n x 1 < x 2 < … < n n x_(1) < x_(2) < dots < n_(n)x_{1}<x_{2}<\ldots<n_{n}x1<x2<…<nnand of n − 1 n − 1 n-1n-1n−1points y 1 , y 2 , … , y n − 1 y 1 , y 2 , … , y n − 1 y_(1),y_(2),dots,y_(n-1)y_{1}, y_{2}, \ldots, y_{n-1}y1,y2,…,yn−1respectively, are consecutive if we can choose the points y i y i y_(i)y_{i}yiso that we have x 1 < y 1 < x 2 < … < y n − 1 < x n x 1 < y 1 < x 2 < … < y n − 1 < x n x_(1) < y_(1) < x_(2) < dots < y_(n-1) < x_(n)x_{1}<y_{1}<x_{2}<\ldots<y_{n-1}<x_{n}x1<y1<x2<…<yn−1<xn.
3. Position of the problem. - The function f ( x ) f ( x ) f(x)f(x)f(x)being a function ( C ) and the points x i x i x_(i)x_{i}xihaving a given distribution, we studied the maximum of the expression
D ( x 1 , x 2 , … , x n ; f ) = f ( x 1 ) , f ( x 2 ) , … , f ( x n ) ∏ i > I 1 , 2 , … , n ( x i − x I ) 2 D x 1 , x 2 , … , x n ; f = f x 1 , f x 2 , … , f x n ∏ i > I 1 , 2 , … , n   x i − x I 2 D(x_(1),x_(2),dots,x_(n);f)=f(x_(1)),f(x_(2)),dots,f(x_(n))prod_(i > j)^(1,2,dots,n)(x_(i)-x_(j))^(2)\mathrm{D}\left(x_{1}, x_{2}, \ldots, x_{n} ; f\right)=f\left(x_{1}\right), f\left(x_{2}\right), \ldots, f\left(x_{n}\right) \prod_{i>j}^{1,2, \ldots, n}\left(x_{i}-x_{j}\right)^{2}D(x1,x2,…,xn;f)=f(x1),f(x2),…,f(xn)∏i>I1,2,…,n(xi−xI)2
  1. Results. - Under these conditions, we found that:
    I. D ( x 1 , x 2 , … , x n ; f ) D x 1 , x 2 , … , x n ; f D(x_(1),x_(2),dots,x_(n);f)\mathrm{D}\left(x_{1}, x_{2}, \ldots, x_{n} ; f\right)D(x1,x2,…,xn;f)has a maximum reached for at least one point system x i x i x_(i)x_{i}xi.
    II. This maximum is reached for a single system of n points, distinct and different from the points a i a i a_(i)a_{i}hasiAnd b i b i b_(i)b_{i}bi. We designate by ξ n , 1 < ξ n , 2 < … < ξ n , n ξ n , 1 < ξ n , 2 < … < ξ n , n xi_(n,1) < xi_(n,2) < dots < xi_(n,n)\xi_{n, 1}<\xi_{n, 2}<\ldots<\xi_{n, n}ξn,1<ξn,2<…<ξn,nthe points for which the maximum is reached. We will say that this is the maximizing system of our problem.
    III. The correspondence between a function ( C ) and its maximizing system is continuous. Therefore, at any ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0, corresponds to a η > 0 η > 0 eta > 0\eta>0η>0such that if g ( x ) g ( x ) g(x)g(x)g(x)is another (C) function verifying | f − g | < η | f − g | < η |f-g| < eta|f-g|<\eta|f−g|<ηon E, we have
| ξ n , i − ξ n , i ′ | < ε ( i = 1 , … , n ) ξ n , i − ξ n , i ′ < ε ( i = 1 , … , n ) |xi_(n,i)-xi_(n,i)^(')| < epsiquad(i=1,dots,n)\left|\xi_{n, i}-\xi_{n, i}^{\prime}\right|<\varepsilon \quad(i=1, \ldots, n)|ξn,i−ξn,i′|<ε(i=1,…,n)
ξ n , i ′ ξ n , i ′ xi_(n,i)^(')\xi_{n, i}^{\prime}ξn,i′being the maximizing system of g ( x ) g ( x ) g(x)g(x)g(x)corresponding to the same problem.
IV. Maximizing systems ξ n , i , ξ n , i ′ ξ n , i , ξ n , i ′ xi_(n,i),xi_(n,i)^(')\xi_{n, i}, \xi_{n, i}^{\prime}ξn,i,ξn,i′corresponding, for the same function f ( x ) f ( x ) f(x)f(x)f(x), to two consecutive distributions of n n nnnpoints, separate. We therefore have
ξ n , 1 < ξ n , 1 ′ < ξ n , 2 < … < ξ n , n < ξ n , n ′ ξ n , 1 < ξ n , 1 ′ < ξ n , 2 < … < ξ n , n < ξ n , n ′ xi_(n,1) < xi_(n,1)^(') < xi_(n,2) < dots < xi_(n,n) < xi_(n,n)^(')quad\xi_{n, 1}<\xi_{n, 1}^{\prime}<\xi_{n, 2}<\ldots<\xi_{n, n}<\xi_{n, n}^{\prime} \quadξn,1<ξn,1′<ξn,2<…<ξn,n<ξn,n′Or ξ n , 1 ′ < ξ n , 1 < ξ n , 2 ′ < … < ξ n , n ′ < ξ n , n ξ n , 1 ′ < ξ n , 1 < ξ n , 2 ′ < … < ξ n , n ′ < ξ n , n quadxi_(n,1)^(') < xi_(n,1) < xi_(n,2)^(') < dots < xi_(n,n)^(') < xi_(n,n)\quad \xi_{n, 1}^{\prime}<\xi_{n, 1}<\xi_{n, 2}^{\prime}<\ldots<\xi_{n, n}^{\prime}<\xi_{n, n}ξn,1′<ξn,1<ξn,2′<…<ξn,n′<ξn,n.
V. Maximizing systems ξ n , i , ξ n − 1 , i ξ n , i , ξ n − 1 , i xi_(n,i),xi_(n-1,i)\xi_{n, i}, \xi_{n-1, i}ξn,i,ξn−1,icorresponding, for the same function f ( x ) f ( x ) f(x)f(x)f(x), at two consecutive distributions of net of n-1 points, separate. We therefore have
ξ n , 1 < ξ n − 1 , 1 < ξ n , 2 < … < ξ n − 1 , n − 1 < ξ n , n . ξ n , 1 < ξ n − 1 , 1 < ξ n , 2 < … < ξ n − 1 , n − 1 < ξ n , n . xi_(n,1) < xi_(n-1,1) < xi_(n,2) < dots < xi_(n-1,n-1) < xi_(n,n).\xi_{n, 1}<\xi_{n-1,1}<\xi_{n, 2}<\ldots<\xi_{n-1, n-1}<\xi_{n, n} .ξn,1<ξn−1,1<ξn,2<…<ξn−1,n−1<ξn,n.
We can say that two maximizing systems of n n nnnpoints or else n n nnnAnd n − 1 n − 1 n-1n-1n−1points always separate, unless this separation is visibly impossible.
5. The Stieltjes problem. - The special case
f ( x ) = ∏ i = 1 m | x − a i | σ i | x − b i | ρ i ( σ i , ρ i positifs ) f ( x ) = ∏ i = 1 m   x − a i σ i x − b i ρ i σ i , ρ i  positifs  f(x)=prod_(i=1)^(m)|x-a_(i)|^(sigma_(i))|x-b_(i)|^(rho_(i))quad(sigma_(i),rho_(i)" positifs ")f(x)=\prod_{i=1}^{m}\left|x-a_{i}\right|^{\sigma_{i}}\left|x-b_{i}\right|^{\rho_{i}} \quad\left(\sigma_{i}, \rho_{i} \text { positifs }\right)f(x)=∏i=1m|x−hasi|σi|x−bi|ρi(σi,ρi positive )
was examined by Th. Stieltjes, who obtained properties I and II in another way ( 1 ) ( 1 ) ^((1)){ }^{(1)}(1)In this case, the polynomial P ( x ) = ∏ i = 1 n ( x − ξ n , i ) P ( x ) = ∏ i = 1 n   x − ξ n , i P(x)=prod_(i=1)^(n)(x-xi_(n,i))\mathrm{P}(x)=\prod_{i=1}^{n}\left(x-\xi_{n, i}\right)P(x)=∏i=1n(x−ξn,i)verifies a linear differential equation of the form
y ′ ′ + ( ∑ i = 1 m σ i x − a i + ρ i x − b i ) y ′ + ψ ( x ) ( x − a 1 ) ( x − b 1 ) … ( x − a m ) ( x − b m ) y = 0 y ′ ′ + ∑ i = 1 m   σ i x − a i + ρ i x − b i y ′ + ψ ( x ) x − a 1 x − b 1 … x − a m x − b m y = 0 y^('')+(sum_(i=1)^(m)(sigma_(i))/(x-a_(i))+(rho_(i))/(x-b_(i)))y^(')+(psi(x))/((x-a_(1))(x-b_(1))dots(x-a_(m))(x-b_(m)))y=0y^{\prime \prime}+\left(\sum_{i=1}^{m} \frac{\sigma_{i}}{x-a_{i}}+\frac{\rho_{i}}{x-b_{i}}\right) y^{\prime}+\frac{\psi(x)}{\left(x-a_{1}\right)\left(x-b_{1}\right) \ldots\left(x-a_{m}\right)\left(x-b_{m}\right)} y=0y′′+(∑i=1mσix−hasi+ρix−bi)y′+ψ(x)(x−has1)(x−b1)…(x−hasm)(x−bm)y=0
ψ ( x ) ψ ( x ) psi(x)\psi(x)ψ(x)being a certain polynomial of degree 2 m − 2 2 m − 2 2m-22 m-22m−2.
We obtained properties IV and V by showing that it is sufficient to demonstrate this for this particular case and even only for certain particular values ​​of the constants σ i σ i sigma_(i)\sigma_{i}σiAnd ρ i ρ i rho_(i)\rho_{i}ρi.
Property IV is a very broad generalization of some results of F. Klein ( 2 2 ^(2){ }^{2}2) and EB van Vleck ( 3 3 ^(3){ }^{3}3). Property V is related, for m = I m = I m=Im=\mathrm{I}m=I, to the theory of Jacobi polynomials ( 4 4 ^(4){ }^{4}4).
  1. ( 1 1 ^(1){ }^{1}1) Acta Mathematica, 6, 1885, p. 321.
    ( 2 2 ^(2){ }^{2}2) Mathematische Annalen, 18, 1881, p. 237.
    ( 3 3 ^(3){ }^{3}3) Bulletin of the Amer. Math. Soc., 2 e 2 e 2^(e)2^{\mathrm{e}}2eseries, 4, 1898, p. 426.
    ( 1 ) ( 1 ) ^((1)){ }^{(1)}(1)Th. J. Stieltjes, Reports, 100, 1885, p. 620.
1936

Related Posts