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Moments and Positivity
Journal of Mathematical Analysis and Applications 247, 570᎐587 Ž2000. doi:10.1006rjmaa.2000.6882, available online at http:rrwww.idealibrary.com on
Moments and Positivity O. Demanze U.F.R. de mathematiques Pures et Appliquees, ´ ´ Uni¨ ersite´ des sciences et Technolgies de Lille, U.R.A. au C.N.R.S. D 751, 59655 Villeneu¨ e d’Ascq Cedex, France E-mail: [email protected] Submitted by Daniel Waterman Received May 27, 1999
We express the structure of some positive polynomials in several variables, as squares of rational function with universal denominators, using functional analysis methods. 䊚 2000 Academic Press
1. INTRODUCTION In the case of one variable, it is well known that every positive polynomial on ⺢ can be written as a sum of squares of polynomials. As Hilbert has already noticed, when we take multi-variable polynomials, this property is false Žsee w1x.. Using algebraic methods, one can prove that every positive polynomial on ⺢ n can be written as a sum of squares of rational fractions, where the denominators depend on the given polynomial Žsee w2, 8x.. In the following, using functional analysis methods, we shall try to find more precise representations of some positive polynomials on ⺢ n, allowing only universal denominators to occur. Specifically, we will show that, under some conditions, such as the positivity of naturally associated polynomials or some properties of the dominating coefficients, we can write strictly positive polynomials on ⺢ n as a sum of squares of rational fractions with denominators of the form
Ž 1 q t12 .
1
⭈⭈⭈ Ž 1 q t n2 .
n
i g ⺪q ,
,
i s 1, . . . , n,
where t s Ž t 1 , . . . , t n . is the variable on ⺢ n. We adapt the techniques from w7x, where the case of homogeneous polynomials has been developed and different universal denominators have been found. Our framework is an algebra of bounded fractions, 570 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
571
MOMENTS AND POSITIVITY
where we can use the properties of a commuting self-adjoint operator family. 1.1 Notation. Let t s Ž t 1 , . . . , t n . be the usual variable of ⺢ n. Then for all multi-indices of ␣ s Ž ␣ 1 , . . . , ␣ n . with ␣ i g ⺪q for all i g 1, . . . , n4 , let t ␣ be t 1␣ 1 ⭈⭈⭈ t n␣ n . In the same way, we write every polynomial P in n variables, PŽ X . s
Ý g␣ X ␣ ,
with X ␣ s X 1␣ 1 ⭈⭈⭈ X n␣ n ,
␣
where the family Ž g ␣ .␣ is almost zero Ži.e., g ␣ s 0 except for a finite n. number of ␣ in ⺪q . We write ␣ F  , if for all i in 1, . . . , n4 , we have ␣ i F i . If ␣ i - i for all i s 1, . . . , n, then we write ␣ -  . For all n multi-indices  s Ž  1 , . . . , n . in ⺪q , we define the functions  on ⺢ n into ⺢ by the relation
 s 1 1 ⭈⭈⭈ n n ,
where i Ž t . s Ž 1 q t i2 .
y1
,
t g ⺢n
2. MOMENTS AND LINEAR POSITIVE SEMI-DEFINITE FUNCTIONALS 2.1 PROPOSITION.
Let the family of rational functions be gi¨ en by t 1 1 t 2 2 ⭈⭈⭈ t n n
Ž 1 q t12 . i g 0, 1 4 ,
m1
⭈⭈⭈ Ž 1 q t n2 .
mn
,
mi G i ,
m i g ⺪q ,
Ž 1. i g 1, . . . , n4 .
These functions form a linearly independent family. Moreo¨ er, if A is the algebra of rational bounded fractions with denominators w  Ž t .xy1 , then the family Ž1. is an algebraic basis of A . Proof. Let us show this by a recurrence on the dimension d. For d s 1, let Ž 1 , . . . , m . g ⺓ m . Suppose that m
Ý i is1
t 1 i
Ž 1 q t12 .
mi
s 0.
Ž 2.
If p s min m i ; i g 1, . . . , m44 , then p may only appear in two fractions: 1
Ž 1 q t12 .
p
or
t1
Ž 1 q t12 .
p
.
572
O. DEMANZE
Let r and Xr be the coefficients multiplying these two functions. In that case, Ž2. implies t 1 i
m
Ý i is1
Ž 1 q t12 .
Ý i
t1ª⬁ is1
s 0,
t 1 i
m
lim
m iyp
Ž 1 q t12 .
m iyp
s 0.
But t 1 i
m
Ý i
lim
t1ª⬁ is1
Ž 1 q t12 .
s r
m iyp
if Xr / 0.
or ⬁
So r and Xr are both equal to zero. Reiterating this process, we obtain that each complex number r is equal to zero, which proves the hypothesis for step 1. Suppose the hypothesis is true up to step d y 1. Let 1 , . . . , l be complex numbers such that t 1 1 , p t 2 2 , p ⭈⭈⭈ t d d , p
l
Ý
p
ps1
Ž 1 q t12 .
m1 ,
p
Ý ps1
⭈⭈⭈ Ž 1 q t d2 .
m d,
p
t 2 2 , p t 3 3 , p ⭈⭈⭈ t d d , p
l
s
p
Ž 1 q t 22 .
m2,
p
⭈⭈⭈ Ž 1 q t d2 .
t 1 1 , p m d,
p
Ž 1 q t12 .
m1 ,
p
s 0.
Applying the hypothesis of step 1, we can write
Ý A , m
p
t 2 2 , p t 3 3 , p ⭈⭈⭈ t d d , p
Ž 1 q t 22 .
m1 ,
p
⭈⭈⭈ Ž 1 q t d2 .
m d,
p
s 0,
where A , m is the set of integers p such that Ž1 q t 12 .ym t 1 appears in the corresponding fractions. These A , m create a partition of 1, . . . , l 4 . This is true for every Ž d y 1.-tuple Ž t 2 , . . . , t d .. Because the recurrence hypothesis is true for d y 1, we obtain that p is null, for all p s 1, . . . , l. 2.2 Remarks. Ž1. Let A be a of complex function algebra whose elements are defined on the Euclidean space ⺢ n, with the following properties: Ži. 1 g A and Žii. if f g A then f g A. Let be a linear positive semi-definite functional on A into ⺓, i.e., Ž ff . G 0, ᭙ f g A. If is not zero, we necessarily have Ž1. ) 0 and we can associate with the pair Ž A, . a pre-Hilbert space. To do that, let N s f g A ; Ž ff . s 0 4 .
Ž 3.
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MOMENTS AND POSITIVITY
By Cauchy᎐Schwarz’s inequality, N is a bilateral ideal. Then, we define the N which is a A-module. In fact, the quotient ArN N may be quotient ArN associated with the following inner product: ² f q N , g q N : s Ž fg . ,
᭙Ž f , g . g A 2 .
For more details, see w4, 5x. Ž2. If A s A then the linear functional is completely determinated, via Proposition 2.1, by its values
aŽm11,, .. .. .. ,, m nn.
s
žŽ
t 1 1 t 2 2 ⭈⭈⭈ t n n 1 q t 12 .
m1
⭈⭈⭈ Ž 1 q t n2 .
mn
/
,
which may be called the moments of . 2.3 DEFINITION. Let ⌽ Ž t 1 , . . . , t n . s w ⌽p, q Ž t 1 , . . . , t n .x1 F p F n, 1 F q F 4 be the map defined on ⺢ n by the following matrix:
⌽ Ž t1 , . . . , t n . s
1
t1
t1
t 12
Ž 1 q t12 .
Ž 1 q t12 .
Ž 1 q t12 .
Ž 1 q t12 .
.. . 1
.. . tn
Ž 1 q t n2 .
Ž 1 q t n2 .
.. . tn
Ž 1 q t n2 .
.. . t n2
.
Ž 1 q t n2 .
The function ⌽ is therefore defined on ⺢ n into Mn= 4 , where Mn=4 is the space of n = 4 matrices. 2.4 LEMMA. The function ⌽ is injecti¨ e on ⺢ n. Proof. The injectivity may be seen because of the following equality. If ⌽ Ž t 1 , . . . , t n . s w x p, q x1 F p F n, 1 F q F 4 , then we can write ti s
xi, 2 xi, 1
,
i g 1, . . . , n4 .
574
O. DEMANZE
Note that we can completely describe the range of ⌽. Specifically ⌽ Ž⺢ n . is the set of all matrices x s w x p, q x1 F p F n, 1 F q F 4 satisfying
¡x
~x
) 0, x i , 2 s x i , 3
᭙ i g 1, . . . , n4 ,
2 i, 1
q x i2, 2 q x i2, 3 q x i2, 4 s 1
᭙ i g 1, . . . , n4 ,
q
᭙ i g 1, . . . , n4 ,
x i2, 1
¢x
¦
i, 1
i, 4
x i2, 2
s xi, 1
¥.
Ž 4.
§
G0
᭙ i g 1, . . . , n4 ,
We define the linear maps B˜p, q on A by the relation A 2 f ª B˜p , q Ž f . s ⌽p , q ⭈ f g A .
Ž 5.
It follows from Ž4. that
Ý
⌽p2, q s n.
Ž 6.
1FpFn , 1FqF4
2.5 Remark. If f is in A , then f may be seen as the composition P ( ⌽ where P is a polynomial in 4 n variables. This representation is clearly not unique. Moreover, we can choose the polynomial P so that it depends only on 3n variables Ždue to the ‘‘symmetry’’ of ⌽ Ž t 1 , . . . , t n ... 2.6 THEOREM. Let be a linear positi¨ e semi-definite functional on A . Then there exist a measure , supported by ⺢ n, and a second one supported by H, included in ⺢ N with N s 4 n, such that for any polynomial P g ⺓ N w X x we ha¨ e
Ž P ( ⌽ . s
H⺢ Ž P ( ⌽ . d q HH P d ,
Ž 7.
n
n where H s Dis1 Hi with
Hi s x s w x p , q x ; x i , 1 s x i , 2 s x i , 3 s 0, x i , 4 s 1, 5 x 5 2 s n ,
½
5
i g 1, . . . , n4 . Proof. Let be as in the statement. We have Ž  < f 2 <. G 0 because for all integers i, ⌽i,2 1 q ⌽i,2 2 s ⌽i, 1 and  s ⌽ 1,11 ⭈ ⌽ 2, 21 ⭈⭈⭈ ⌽n,1n . Let H be the Hilbert space associated with the linear positive functional Žsee Remark 2.2.. We can define Ž Bp, q . on H in the same way as we have defined Ž B˜p, q . on A . The operators Bp, q become bounded
575
MOMENTS AND POSITIVITY
commuting self-adjoint operators verifying Bp2, 1 q Bp2, 2 q Bp2, 3 q Bp2, 4 s I, Bp2, 1 q Bp2, 2 s Bp , 1 , Bp , 2 s Bp , 3 , yI F Bp , q F I, 0 F Bp , q F I,
p s 1, . . . , n,
p s 1, . . . , n,
Ž 8. Ž 9.
p s 1, . . . , n,
Ž 10 .
for Ž p, q . such that 1 F p F n, 2 F q F 3, Ž 11 . for Ž p, q . such that 1 F p F n,
Ý
q g 1, 4 4 , Ž 12 .
Bp2, q s nI.
Ž 13 .
1FpFn , 1FqF4
Properties Ž8., Ž9., and Ž10. come from corresponding properties Ž4. of ⌽ Ž⺢ n .. For the remaining properties, take two elements f and g of A , N defined by f q N and g q N , and denote by f˜and ˜ g the elements of ArN respectively. Then, ² Bp , q f˜, ˜ g : s Ž ⌽p , q f ⭈ g . s Ž f ⭈ ⌽p , q g . s ² f˜, Bp , q ˜ g : . So all the operators Bp, q are symmetric. To end the proof of the properties, we may assume p equals 1 Žfor the others operators, the proof is the same because of the ‘‘symmetry’’ of the operator family.. As we have 2 2 2 2 ⌽ 1, 1 q ⌽ 1, 2 q ⌽ 1, 3 q ⌽ 1, 4 s 1,
using the linearity of and the fact that each ⌽ 1, j is real, we obtain the following equality: 5 f˜5 2 s 5 ⌽ 1, 1 f q N 5 2 q 5 ⌽ 1, 2 f q N 5 2 q 5 ⌽ 1, 3 f q N 5 2 q 5 ⌽ 1, 4 f q N 5 2 , 5 f˜5 2 s 5 B1, 1 f˜5 2 q 5 B1, 2 f˜5 2 q 5 B1, 3 f˜5 2 q 5 B1, 4 f˜5 2 . We deduce that B1, q for q in 1, . . . , 44 is a contraction, and so a bounded operator. Now we just have to show relations Ž11., Ž12., and Ž13.. As we know that ⌽ 1, 1 s ⌽ 1,2 1 q ⌽ 1,2 2 , one more time, due to the linearity of , 2 2 ² B1, 1 f˜, f˜: s Ž ⌽ 1, 1 ff . s Ž ⌽ 1, 1 ff . q Ž ⌽ 1, 2 ff .
s 5 B1, 1 f˜5 2 q 5 B1, 2 f˜5 2 G 0.
576
O. DEMANZE
Moreover, as B1, 1 is a contractive self-adjoint operator, 0 F B1, 1 F I. For B1, 4 it is exactly the same result because ⌽ 1, 4 s ⌽ 1,2 3 q ⌽ 1,2 4 . Due to the equality Ž8., we now prove that B1, 2 and B1, 3 satisfy B1,2 2 q B12, 3 F I. And so, yI F B1 , 2 s B1, 3 F I. Finally, for Ž13., we just have to use Ž6.. As all these identities and N , which is a dense subspace of H , these inequalities are true for ArN relations are also true on H. We will denote by B the self-adjoint bounded operator family Ž Bp, q .1 F p F n, 1 F q F 4 . Let Ž B . be the joint spectrum of the self-adjoint bounded operator family B. Ž B . is a subset of Ž⺢ 4 . n. So, due to the Gelfand theory, each element of the joint spectrum is either included in ⌽ Ž⺢ n . or in at least one Hi . Let x g Ž B . and let ␥ be a character associated to x. We denote by Ž␥ i, j .1 F iF n, 1 F jF 4 the family Ž␥ Ž Bi, j ..1 F iF n, 1 F jF 4 . Using properties Ž8., Ž9., Ž10., and Ž12., we verify that x s Ž␥ i, j .1 F iF n, 1 F jF 4 is in ⌽ Ž⺢ n . if each ␥ i, 1 is non-zero; if x is not in ⌽ Ž⺢ n ., there exists an element i 0 in 1, . . . , n4 so that ␥ i 0 , 1 s 0. Then by Ž9. ␥ i , 2 s ␥ i , 3 s 0 and using the relations Ž8. and Ž12. we have that 0 0 ␥ i 0 , 4 s 1. So x is necessarily in Hi 0 . Thus we have the following inclusion:
Ž B. : ⌽Ž⺢n. j H,
⌽ Ž ⺢ n . l H s ⭋.
Ž 14 .
We define now
0 s Ž B . l H 1 s Ž B . l ⌽ Ž ⺢ n . . Let E be the joint spectral measure of B; 0 and 1 are two disjoint Borel sets of ⺢ N. Then for all Borel sets of ⺢ N, we define the following positive measures:
Ž t . s ² E Ž l 0 . 1, 1: Ž . s ² E Ž l 1 . 1, 1: . We construct a third one, say , by the relation
Ž X . s Ž ⌽Ž X . . ,
᭙ X Borel sets of ⺢ n .
MOMENTS AND POSITIVITY
577
is a measure because ⌽ is bijective from ⺢ n onto ⌽ Ž⺢ n ., so we can write Ž X . s w ⌽y1 x
ž
y1
Ž X . / .
We have
Ž P ( ⌽ . s ² P Ž B . .1, 1: s
H⺢
N
P.d² E.1, 1: .
Moreover, ² E Ž . 1, 1: s ² E Ž l 0 . 1, 1: q ² E Ž l 1 . 1, 1: . Therefore,
Ž P ( ⌽ . s
H⌽ Ž⺢ .P d q H⺢ n
N
P d .
As is supported by H and by ⌽ Ž⺢ n .,
Ž P ( ⌽ . s
H⌽ Ž⺢ . P d q HH P d , n
H⌽ Ž⺢ . P d s H⺢ n
n
Ž P (⌽. d.
Consequently,
Ž P ( ⌽ . s
H⺢
n
Ž P ( ⌽ . d q H P d . H
2.7 COROLLARY. Let Ž 1 , . . . , m . be an m-tuple of real elements of A . Let Tj be polynomials in 4 n ¨ ariables such that j s Tj ( ⌽, for j in 1, . . . , m4 . Let be a linear positi¨ e semi-definite functional such that Ž j < f 2 <. G 0, ᭙ j g 1, . . . , m4 . Assume that P is in ⺓ N w X x. Then we can find two positi¨ e measures and , supported by Fj ŽTj ( ⌽ .y1 Ž⺢q . and Fj Tjy1 Ž⺢q ., respecti¨ ely, such that
Ž P ( ⌽ . s
HI Ž P ( ⌽ . d q HH P d , X
where I s Fj ŽTj ( .y1 Ž⺢q . and H X s H l Fj Tjy1 Ž⺢q ..
Ž 15 .
578
O. DEMANZE
Proof. As Ž j < f 2 <. is positive for all integers j g 1, . . . , m4 , we obtain the condition Tj Ž B . positive for all integers j because ²Tj Ž B . f˜, f˜: s ²Tj ( ⌽ f˜, f˜: s ² j f˜, f˜: s j < f < 2 G 0.
ž
/
Let ␥ be a character; hence Tj Ž␥ Ž B .. s ␥ ŽTj Ž B .. G 0. We deduce that ␥ Ž B . g Tjy1 Ž⺢q .. Consequently, the spectral measure associated with the joint spectrum Ž B . has its support included in Fj Tjy1 Ž⺢q .. 3. POLYNOMIAL REPRESENTATIONS In this section, we shall use the representation of a linear positive semi-definite functional on A into the sum of two integrals with respect to two positive measures to describe the structure of some positive polynomials. 3.1 LEMMA. Let S be a real linear space, and let C be a positi¨ e con¨ ex cone included in S with the property S s C y C . Assume that C can be written as Dd G 1 Cd where the Cd is a con¨ ex cone such that the Sd s Cd y Cd are linear spaces of finite dimension, and such that Cdq 1 l Sd s Cd for all strictly positi¨ e integers d. Denote by intŽ Cd . the relati¨ e interior of Cd as a subset of the Euclidean space Sd . Assume that there exists an element g C1 with the property that, for all strictly positi¨ e d and for all linear non-zero positi¨ e functionals in SdU on Cd , one has Ž . ) 0. Let r 0 g Sd 0 y intŽ Cd 0 ., where d 0 is a strictly positi¨ e integer; then there exists a linear functional defined on S into ⺢ such that Ž r 0 . F 0 and is strictly positi¨ e on intŽ Cd . for all d more than d 0 . In particular restricted to C is positi¨ e. We can find the proof of the above lemma in w7x Žwhich, in turn, uses a method due to Cassier; see w3x.. 3.2 DEFINITION.
For all positive integers d, we define
Fd s g Ž t . s
½
Ý a␣ ,  t ␣ Ž t . 
␣, 
with ␣ F 2  F 4 D ,
5
D s Ž d, d, . . . , d . . If P is a polynomial defined on ⺢ n, we denote by ␦ j Ž P . the degree of P as a one variable polynomial in X j on the Euclidean ring ⺢w X 1 , . . . , X jy1 ,
579
MOMENTS AND POSITIVITY
X jq1 , . . . , X n x. Then, we denote by ␦ Ž P . the ‘‘multi-degree’’ of P, i.e., the following multi-index:
␦ Ž P . s ␦ 1 Ž P . , . . . , ␦n Ž P . 4 .
Ž 16 .
Let C be the positive cone generated by the elements of the form r 2 , r g A and Pj ␥ Ž P j . s 2 , s g A , where P1 , . . . , Pk are fixed real polynomials. In the same way, let ⌺ be the positive cone included in C , generated by the elements of the form r 2 , r g A . Let P Ž X . s Ý ␣ g ␣ X ␣ be a polynomial in n variables of even multi-degree. We can associate P with an other polynomial P˜ in 3n variables via the formula P˜Ž X , Y , Z . s
Ý g␣ X  Y ␣y2  Z ␦ Ž P .r2q y ␣ ,
Ž 17 .
␣
or equivalently, P˜Ž X , Y , Z . s Z ␦ Ž P .r2
Ý g␣ ␣
X

Y
␣ y2 
ž /ž / Z
Z
,
Ž 18 .
where  is the multi-index Žw ␣ 1r2x, . . . , w ␣ nr2x. Žw.x is the integer part function.. This polynomial P˜ will be composed on the right with the function ⌽ Žregarding P˜ as a 4 n variable polynomial.. To understand this representation, it is easier to index again the different variables. Recalling that every element of ⌽ Ž⺢ n . can be expressed by a matrix x s w x p, q x1 F p F n, 1 F q F 4 , let us write P˜ in the following form: P˜Ž X 1, 1 , . . . , X n , 4 . s
Ý g␣ X1, 4 1
␣
⭈⭈⭈ X n,n4 X 1␣, 12y2  1 ⭈⭈⭈
X n␣,n2y2  n X 1␦ 1Ž P .r2q  1y␣ 1 ⭈⭈⭈ X n␦,n1Ž P .r2q  ny␣ n . Ž 19 . This means that the n variables X, Y, and Z play the role of Ž ⌽ 1, 4Ž t ., . . . , ⌽n, 4 Ž t .., Ž ⌽ 1, 2 Ž t ., . . . , ⌽n, 2 Ž t .., and Ž ⌽ 1, 1Ž t ., . . . , ⌽n, 1Ž t .., respectively. An easy calculation gives the relations P ␦ Ž P .r2 Ž t . s P˜( ⌽ Ž t . , P˜Ž tX , t , 1 . s P Ž t . ,
᭙t g ⺢ n. ᭙t g ⺢n,
Ž 20 . Ž 21 .
where tX s Ž t 12 , . . . , t n2 .. Formula Ž19. gives an explicit way to find the polynomial P˜ associated with the polynomial P.
580
O. DEMANZE
3.3 THEOREM. Let Ž P1 , . . . , Pk . be a k-tuple of polynomials of multi-degrees 2 D 1 , . . . , 2 D k , respecti¨ ely, and let P be an other polynomial, strictly k positi¨ e on Fis1 Piy1 Ž⺢q ._ 04 , of multi-degree 2 D. Let P˜1 , . . . , P˜k , P˜ be their associated polynomials. Assume that P˜Ž t . ) 0 for each t in H l k Fis1 P˜iy1 Ž⺢q . where H has been defined in Theorem 2.6. Then there exist a positi¨ e multi-index M and a finite number of real polynomials ql 4l g L and qi, l 4i g 1, . . . , k4, l g LX such that PŽ t. s Ž t.
2M
k
ž
Ý ql2 Ž t . q Ý Ý Pi Ž t . qi2, l Ž t . X
lgL
is1 lgL
t g ⺢n,
/
,
n M g ⺪q .
Ž 22 .
Proof. We want to apply Lemma 3.1, so we begin to verify the conditions of this lemma. Set Cd s C l Fd and Sd s Cd y Cd , which is a linear space. As Cd ; Sd ; Fd , Sd is a finite dimensional space. Moreover, A s C y C , because every element f in S can be written 14 Žw f q 1x 2 y w f y 1x 2 .. We have
Ž Cdq 1 l Sd . ; Ž Cdq1 l Fd . s Ž C l Fdq1 l Fd . s Ž C l Fd . s Cd . Due to the three following inclusions Ž Cdq 1 l Sd . ; Cd , Cd ; Cdq1 , Cd ; Sd , we have Ž Cdq1 l Sd . s Cd . We must find now an element g C1 such that, for all strictly positive d and for every non-zero linear functional in SdU which is positive on Cd , we have Ž . ) 0. Choose s 1, fix an integer d, let in SdU , assume that Ž . s 0, and let us prove that is zero in Sd . It suffices to verify that s 0 on C . Let F 2 D Ž t . s Ž1 q t 12 . 2 d ⭈⭈⭈ Ž1 q t n2 . 2 d ; then F 2 D Ž t . is a sum of squares where t 2 ␣ s t 12 ␣ 1 ⭈⭈⭈ t n2 ␣ n appears with ␣ i less than 2 d for every i s 1, . . . , n. So, 1y
t2␣ F2DŽ t.
g Cd ,
᭙␣ F 2 D.
As Ž1. s 0, then y w t 2 ␣rF 2 D Ž t .x G 0, but w t 2 ␣rF 2 D Ž t .x s wŽ t ␣r F D Ž t .. 2 x G 0. Thus,
t2␣ F2DŽ t.
s 0,
᭙␣ F 2 D.
Ž 23 .
581
MOMENTS AND POSITIVITY
Let ␣ F 2 D and  F 2 D and let g ⺢; then t ␣rF D Ž t . and t rF D Ž t . are in Fd , so
ž
t␣
q
F DŽ t. s 2
2
t F DŽ t.
t2 F2DŽ t.
/
q 2
t ␣q  F2DŽ t.
t2␣
q
F2DŽ t.
G 0.
According to Ž23., t2
F2DŽ t.
t2␣
s
s 0.
F2DŽ t.
Since for every real number , w t ␣q rF 2 D Ž t .x is positive, one has
t ␣q 
᭙␣ F 2 D,
s 0,
F2DŽ t.
᭙ F 2 D.
Ž 24 .
If ␣ F 4 D, write ␣ s ␣ 1 q ␣ 2 with ␣ 1 F 2 D and ␣ 2 F 2 D. Applying the above result with ␣ 1 and ␣ 2 we obtain
t ␣ 1q ␣ 2
s
F2DŽ t.
t␣ F2DŽ t.
᭙␣ F 4 D.
s 0,
Ž 25 .
We have just to see that w t ␣rF  Ž t .x s 0 when ␣ F 2  F 4 D. Let i g 1, . . . , n4 , and let i s 2 k i q i , with i g 0, 14 . Then t␣ F Ž t. t␣ FŽ t.
t i␣ i
n
s
Ł
is1
Fi2 k iq i Ž t . t i␣ i
n
s
Ł
is1, is0
where Fi Ž t . s 1 q t i2 ,
,
t i␣ i
n
Fi2 k iq i
.
Ł
is1, is1
Fi2 k iq i Ž t .
.
Let I s i such that i s 04 and I X s i such that i s 14 . Also let Ž e1 , . . . , e n . be the canonical basis of ⺢ n. We have two cases. Ža. If i g I, then ␣ F 2  F 4 D becomes ␣ i F 4 k i F 4 d, t i␣ i F Ž2 k i . e i Ž t .
s
t i␣ i FiŽ2 dy2 k i . e i Ž t . F Ž2 d. e i Ž t .
,
582
O. DEMANZE
with the degree of the numerator equal to ␣ i q 2Ž2 d y 2 k i ., which is less than 4 d. Žb. If i g I X , then ␣ F 2  F 4 D becomes ␣ i F 4 k i q 2 F 4 d, t i␣ i F Ž2 k iq1. e i Ž t .
s
t i␣ i F e i Ž t . F Ž2 k iq2. e i Ž t .
.
As 2 k i q 1 F 2 d, 2 k i q 2 F 2 d, we have t i␣ i F Ž2 k iq1. e i Ž t .
s
t i␣ i F e i F Ž2 dy2 k iy2 . e i Ž t . F Ž2 d. e i Ž t .
,
with the degree of the numerator equal to ␣ i q 2 q 2.Ž2 d y 2 k i y 2., which is less than 4 d too. Writing that t␣ FŽ t.
s
t ␣ F 2 Dy  Ž t . F2DŽ t.
,
when ␣ F 2  F 4 D, we can now use Ž25.. We conclude that
t␣ FŽ t.
s0
when ␣ F 2  F 4 D.
Ž 26 .
So s 0 in Fd , and we can apply Lemma 3.1. Clearly, P ␦ Ž P .r2 g Fd s Sd . Assume that P ␦ Ž P .r2 f IntŽ Cd .; then there exists a linear functional ⌿ in A into ⺓ such that its restriction to C is positive and ⌿ Ž P ␦ Ž P .r2 . F 0. Because of the positivity of ⌿ on C , we can apply Corollary 2.7. So there exist two positive measures and such that ⌿ Ž P ␦ Ž P .r2 . s
HI P˜( ⌽ d q HH P˜d , X
where P˜( ⌽ s P ␦ Ž P .r2 , I s Fj Pjy1 Ž⺢q ., and H X s H l Fj P˜jy1 Ž⺢q .. Assuming that has an empty support, then the spectrum Ž B . is in the range by ⌽ of the support suppŽ . of , which is an nonempty set. But we have assumed that P was strictly positive in I, so it is the same for P ␦ Ž P .r2 which is P˜( ⌽. In this case HI P˜( ⌽ d ) 0. If has a nonempty support, as we have assumed that P˜ ) 0 in suppŽ ., HH X P˜d ) 0. The two
583
MOMENTS AND POSITIVITY
measures cannot have simultaneously an empty support because Ž B . is a nonempty set, and so in every case 0 G ⌿ Ž P ␦ Ž P .r2 . s
H⺢
n
P˜( ⌽ d q
HH P˜d ) 0.
But this is impossible, so P ␦ Ž P .r2 is in the interior of Cd , and in particular in Cd . Hence there exist a finite number of elements in A ql 4l g L and qi, l 4i g 1, . . . , k4, l g LX such that PŽ t. Ž t.
␦ Ž P .r2
k
s
Ž . Ý ql2 Ž t . q Ý Ý Pi Ž t . Ž t . ␦ P qi2, l Ž t . , i
X
lgL
t g ⺢n;
is1 lgL
that is, k
PŽ t. s
X
Ý
l gL 1
q1,2 lX Ž t . q
Ý Ý is1 l
X
X gL 1
Pi Ž t . Ž t .
␦ Ž Pi .
q1,2 i , lX Ž t . ,
t g ⺢n,
where q1, lX 4lX g L 1 and q1, i, lX 4i g 1, . . . , k4, lX g LX1 are not necessarily in A , but n. their denominators are of the form w Ž t . F xy1 Ž F g ⺪q . Taking the same denominator, because ␦ Ž Pi . s 2 Di for all integers i, we derive PŽ t. s Ž t.
2M
k
ž
Ý
X
X
l gL
r l2X Ž t . q
Ý Ý Pi Ž t . ri2, l Ž t . X
X
X
is1 l gL
/
,
t g ⺢n,
where r lX , ri, lX 4lX g LX , i g 1, . . . , k4 are polynomials in ⺢ nw X x. 3.4 COROLLARY. If P is a polynomial in n ¨ ariables such that P˜ is strictly n positi¨ e in ⺢ N _ 04 , then there exists M in ⺪q such that PŽ t. s Ž t.
2M
t g ⺢n,
2 l
½ Ý q Ž t. 5 , lgL
where ql 4l g L are polynomials in ⺢ nw X x. Proof. As P˜ is a strictly positive polynomial on ⺢ N _ 04 , P is strictly positive on ⺢ n because of Ž21.. So the multi-degree of P is even. We can apply the above result with all polynomials Pj equal to 0. 3.5 THEOREM.
Let P be a strictly positi¨ e polynomial on ⺢ n such that PŽ t. s
ž
Ý
g 0, 1 4 n _ 0 4
b M t 2 M q Q Ž t . ,
/
584
O. DEMANZE
where b M ) 0 for all . If 2 M s Ž2 m1 , . . . , 2 m n . is the multi-degree of P, n assume that ␦ Ž Q . - 2 M. Then there exists N in ⺪q such that PŽ t. s 2NŽ t.
2 l
½ Ý q Ž t. 5 ,
t g ⺢n,
lgL
where Ž ql . l g L are polynomials in ⺢ nw X x. Proof. Let ␦ Ž P . be the multi-degree of P, ␦ Ž P . s Ž2 m1 , . . . , 2 m n .. Then P Ž t . ␦ Ž P .r2 Ž t . is in A . Let ⌺ be defined as in the beginning of the section. As in Theorem 3.3, one can use Lemma 3.1 due to w7x, letting ⌺ d s Fd l ⌺. Assume that P Ž t . ␦ Ž P .r2 Ž t . f IntŽ ⌺ d .; then there exists a linear functional ⌿ on A into ⺓ such that the restriction of ⌿ to ⌺ is positive and such that ⌿ Ž P ␦ Ž P .r2 . F 0. Because of the positivity of ⌿ in ⌺, Theorem 2.6 can be applied. Hence there exist two positive measures and such that ⌿ Ž P ␦ Ž P .r2 . s
H⺢
n
s
H⺢
n
P˜( ⌽ d q
HH P˜d
P Ž t . ␦ Ž P .r2 Ž t . d Ž t . q
HH P˜d .
If suppŽ . / ⭋, then H⺢ n P Ž t . ␦ Ž P .r2 Ž t . d Ž t . ) 0. If suppŽ . s ⭋, then suppŽ . / ⭋, and let g suppŽ ., g Ž⺢ 4 . n. Then there exists an integer j0 in 1, . . . , n4 such that
Ž j , 1 , j , 2 , j , 3 , j , 4 . s Ž 0, 0, 0, 1. 0
0
0
0
Ždecomposition of Ž B . into two disjoint parts included respectively in ⌽ Ž⺢ n . and in H .. Via a change of indices, we may assume j0 s 1. Let M X be the multi-index Ž m 2 , . . . , m n .. We have P Ž t . ␦ Ž P .r2 Ž t . s
Ý
g 0, 1 _ 0 4 4n
b M t 2 M ␦ Ž P .r2 Ž t .
q ␦ Ž P .r2 Ž t . Q Ž t . . s
Ý
Xg 0, 1 4 ny1 _ 0 4
q
Ý
Xg 0, 1 4 ny1
b 0, X M X t Ž0, 2
X
X
M . ␦ Ž P .r2
bm 1 , X M X t 12 m 1 t Ž0, 2
q ␦ Ž P .r2 Ž t . Q Ž t . .
X
X
Ž t.
M . ␦ Ž P .r2
Ž t.
585
MOMENTS AND POSITIVITY
So, using formula Ž19., we obtain P˜Ž x . s
Ý
Xg 0, 1 4 ny1 _ 0 4
q
Ý
Xg 0, 1 4 ny1
b 0, X M X Ž x 1, 1 .
bm 1 , X M X Ž x 1, 4 .
m1
Ž ˆx 1 .
m1
XM X
Ž ˆx 1 .
XM X
q x 1, 1 Q1 Ž x . q x 1, 2 Q2 Ž x . , where Ž ˆ x1.
X
M
X
j.m j s Ł njs2 x j, j4m j x Ž1y . As Ž 1, 1 , 1, 2 , 1, 3 , 1, 4 . s Ž0, 0, 0, 1., j, 1
P˜Ž . s
Ý
Xg 0, 1 4 ny1 X
bm 1 , X M X Ž ˆ1 .
XM X
,
X
j.m j where Ž ˆ1 . M s Ł njs2 j,4j m j j,Ž1y . 1 X Let b s minw bm 1 , X M X , g 0, 14 ny 1 x. Because every j, 4 and every j, 1 is positive
P˜Ž . G b
Ý
Xg 0, 1 4 ny1
Ž ˆ1 .
XM X
n
sb X
.m . Ý j,4m j,Ž1y 1 j
Ý
j
j
j
g 0, 1 4 ny1 js2 j.m j As in the sum Ý X g 0, 14ny 1 Ł njs2 j,4j m j j,Ž1y every element of the form 1 mj n Ł js2 j, *j appears, where *j take the values 1 and 4, we obtain the following equality:
n
X
Ý
j.m j s Ł j,4j m j j,Ž1y 1
g 0, 1 4 ny1 js2
n
Ł Ž j,m1 q j,m4 . . j
j
js2
Therefore, n
P˜Ž . G b Ł Ž j,m1 j q j,m4 j . . js2
Since j, 1 and j, 4 are positive and cannot be zero simultaneously, for all j Žas is in the spectrum of B, using Ž8., Ž9., and Ž12., we know that if j, 1 s 0 then j, 4 s 1., one obtains that P˜Ž . ) 0; thus HH P˜d ) 0. The two supports cannot be empty simultaneously, so 0 G ⌿ Ž P ␦ Ž P .r2 . s
H⺢
n
P˜( ⌽ d q
HH P˜d ) 0,
586
O. DEMANZE
where P˜( ⌽ s P ␦ Ž P .r2 . That is impossible; thus P ␦ Ž P .r2 is in the interior of one ⌺ d and, in particular, in ⌺ d . So there exists a finite number of elements of A , ql 4l g L , such that PŽ t. Ž t.
␦ Ž P .r2
s
Ý ql2 Ž t . ,
t g ⺢ n.
lgL
Consequently, there exists a finite number of elements of ⺢ nw X x, qXlX 4lX g LX , such that PŽ t. s 2NŽ t.
Ý
X
X
l gL
qXlX2 Ž t . ,
t g ⺢ n.
Theorem 3.5 gives a representation of strictly positive polynomials on ⺢ n, where we do not assume any condition on the associated polynomial ˜ This result can be seen as a result similar to the Artin theorem which P. gives an answer to the 17th Hilbert problem ŽParis 1900.; see w2, 6x. 3.6 EXAMPLE. It is known that the polynomial x 12 x 22 Ž x 12 q x 22 y 1. q 1 is strictly positive on ⺢ 2 Žbut it cannot be written as a sum of squares of polynomials; see w1x.. Let P Ž x 1 , x 2 . s x 16 x 26 q x 16 q x 26 q x 12 x 22 Ž x 12 q x 22 y 1 . q 1. An easy calculation gives P˜Ž X , Y , Z . s X 13 X 23 q X 13 Z23 q Z13 X 23 q X 12 Z1 X 2 Z22 q X 1 Z12 X 22 Z2 y X 1 Z12 X 2 Z22 q Z13 Z23 . In particular, P˜Ž X, Y, 0. s X 13 X 23, so P˜ does not satisfy the conditions of Corollary 3.4. Nevertheless, using Theorem 3.5, P is in the positive cone ⌺; i.e., it can be written as a sum of squares of fractions where the 2 denominators of these fractions are of the form of w F xy1 with F in ⺪q . REFERENCES 1. C. Berg, J. P. R. Christensen, and P. Ressel, ‘‘Harmonic Analysis on Semigroups,’’ Springer-Verlag, New YorkrBerlinrHeidelbergrTokyo, 1984. 2. J. Bochnak, M. Coste, and M. M. Roy, ‘‘Geometrie algebrique reelle,’’ Springer-Verlag, ´ ´ ` ´ BerlinrHeidelbergrNew YorkrLondonrParisrTokyo, 1987. 3. G. Cassier, Probleme de polynomes ` des moments sur un compact de ⺢ n et decomposition ´ ˆ `a plusieurs variables, J. Funct. Anal. 58 Ž1984., 254᎐266. 4. N. Dunford and J. T. Schwartz, ‘‘Linear Operators, Part II,’’ Interscience, New Yorkr LondonrSydney, 1963.
MOMENTS AND POSITIVITY
587
5. B. Fuglede, The multidimensional moment problem, Exposition. Math. 1 Ž1983., 47᎐65. 6. N. Jacobson, ‘‘Theory of Fields and Galois Theory,’’ Lectures in Abstract Algebra III, Van Nostrand, TorontorNew YorkrLondon, 1975. 7. M. Putinar and F. H. Vasilescu, Solving moment problems by dimensional extension, Ann. Math. 149 Ž1999., 1087᎐1107. 8. I. R. Shafarevich, ‘‘Basic Algebraic Geometry. 1,’’ Springer-Verlag, Berlin, 1994.
Moments and Positivity O. Demanze U.F.R. de mathematiques Pures et Appliquees, ´ ´ Uni¨ ersite´ des sciences et Technolgies de Lille, U.R.A. au C.N.R.S. D 751, 59655 Villeneu¨ e d’Ascq Cedex, France E-mail: [email protected] Submitted by Daniel Waterman Received May 27, 1999
We express the structure of some positive polynomials in several variables, as squares of rational function with universal denominators, using functional analysis methods. 䊚 2000 Academic Press
1. INTRODUCTION In the case of one variable, it is well known that every positive polynomial on ⺢ can be written as a sum of squares of polynomials. As Hilbert has already noticed, when we take multi-variable polynomials, this property is false Žsee w1x.. Using algebraic methods, one can prove that every positive polynomial on ⺢ n can be written as a sum of squares of rational fractions, where the denominators depend on the given polynomial Žsee w2, 8x.. In the following, using functional analysis methods, we shall try to find more precise representations of some positive polynomials on ⺢ n, allowing only universal denominators to occur. Specifically, we will show that, under some conditions, such as the positivity of naturally associated polynomials or some properties of the dominating coefficients, we can write strictly positive polynomials on ⺢ n as a sum of squares of rational fractions with denominators of the form
Ž 1 q t12 .
1
⭈⭈⭈ Ž 1 q t n2 .
n
i g ⺪q ,
,
i s 1, . . . , n,
where t s Ž t 1 , . . . , t n . is the variable on ⺢ n. We adapt the techniques from w7x, where the case of homogeneous polynomials has been developed and different universal denominators have been found. Our framework is an algebra of bounded fractions, 570 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
571
MOMENTS AND POSITIVITY
where we can use the properties of a commuting self-adjoint operator family. 1.1 Notation. Let t s Ž t 1 , . . . , t n . be the usual variable of ⺢ n. Then for all multi-indices of ␣ s Ž ␣ 1 , . . . , ␣ n . with ␣ i g ⺪q for all i g 1, . . . , n4 , let t ␣ be t 1␣ 1 ⭈⭈⭈ t n␣ n . In the same way, we write every polynomial P in n variables, PŽ X . s
Ý g␣ X ␣ ,
with X ␣ s X 1␣ 1 ⭈⭈⭈ X n␣ n ,
␣
where the family Ž g ␣ .␣ is almost zero Ži.e., g ␣ s 0 except for a finite n. number of ␣ in ⺪q . We write ␣ F  , if for all i in 1, . . . , n4 , we have ␣ i F i . If ␣ i - i for all i s 1, . . . , n, then we write ␣ -  . For all n multi-indices  s Ž  1 , . . . , n . in ⺪q , we define the functions  on ⺢ n into ⺢ by the relation
 s 1 1 ⭈⭈⭈ n n ,
where i Ž t . s Ž 1 q t i2 .
y1
,
t g ⺢n
2. MOMENTS AND LINEAR POSITIVE SEMI-DEFINITE FUNCTIONALS 2.1 PROPOSITION.
Let the family of rational functions be gi¨ en by t 1 1 t 2 2 ⭈⭈⭈ t n n
Ž 1 q t12 . i g 0, 1 4 ,
m1
⭈⭈⭈ Ž 1 q t n2 .
mn
,
mi G i ,
m i g ⺪q ,
Ž 1. i g 1, . . . , n4 .
These functions form a linearly independent family. Moreo¨ er, if A is the algebra of rational bounded fractions with denominators w  Ž t .xy1 , then the family Ž1. is an algebraic basis of A . Proof. Let us show this by a recurrence on the dimension d. For d s 1, let Ž 1 , . . . , m . g ⺓ m . Suppose that m
Ý i is1
t 1 i
Ž 1 q t12 .
mi
s 0.
Ž 2.
If p s min m i ; i g 1, . . . , m44 , then p may only appear in two fractions: 1
Ž 1 q t12 .
p
or
t1
Ž 1 q t12 .
p
.
572
O. DEMANZE
Let r and Xr be the coefficients multiplying these two functions. In that case, Ž2. implies t 1 i
m
Ý i is1
Ž 1 q t12 .
Ý i
t1ª⬁ is1
s 0,
t 1 i
m
lim
m iyp
Ž 1 q t12 .
m iyp
s 0.
But t 1 i
m
Ý i
lim
t1ª⬁ is1
Ž 1 q t12 .
s r
m iyp
if Xr / 0.
or ⬁
So r and Xr are both equal to zero. Reiterating this process, we obtain that each complex number r is equal to zero, which proves the hypothesis for step 1. Suppose the hypothesis is true up to step d y 1. Let 1 , . . . , l be complex numbers such that t 1 1 , p t 2 2 , p ⭈⭈⭈ t d d , p
l
Ý
p
ps1
Ž 1 q t12 .
m1 ,
p
Ý ps1
⭈⭈⭈ Ž 1 q t d2 .
m d,
p
t 2 2 , p t 3 3 , p ⭈⭈⭈ t d d , p
l
s
p
Ž 1 q t 22 .
m2,
p
⭈⭈⭈ Ž 1 q t d2 .
t 1 1 , p m d,
p
Ž 1 q t12 .
m1 ,
p
s 0.
Applying the hypothesis of step 1, we can write
Ý A , m
p
t 2 2 , p t 3 3 , p ⭈⭈⭈ t d d , p
Ž 1 q t 22 .
m1 ,
p
⭈⭈⭈ Ž 1 q t d2 .
m d,
p
s 0,
where A , m is the set of integers p such that Ž1 q t 12 .ym t 1 appears in the corresponding fractions. These A , m create a partition of 1, . . . , l 4 . This is true for every Ž d y 1.-tuple Ž t 2 , . . . , t d .. Because the recurrence hypothesis is true for d y 1, we obtain that p is null, for all p s 1, . . . , l. 2.2 Remarks. Ž1. Let A be a of complex function algebra whose elements are defined on the Euclidean space ⺢ n, with the following properties: Ži. 1 g A and Žii. if f g A then f g A. Let be a linear positive semi-definite functional on A into ⺓, i.e., Ž ff . G 0, ᭙ f g A. If is not zero, we necessarily have Ž1. ) 0 and we can associate with the pair Ž A, . a pre-Hilbert space. To do that, let N s f g A ; Ž ff . s 0 4 .
Ž 3.
573
MOMENTS AND POSITIVITY
By Cauchy᎐Schwarz’s inequality, N is a bilateral ideal. Then, we define the N which is a A-module. In fact, the quotient ArN N may be quotient ArN associated with the following inner product: ² f q N , g q N : s Ž fg . ,
᭙Ž f , g . g A 2 .
For more details, see w4, 5x. Ž2. If A s A then the linear functional is completely determinated, via Proposition 2.1, by its values
aŽm11,, .. .. .. ,, m nn.
s
žŽ
t 1 1 t 2 2 ⭈⭈⭈ t n n 1 q t 12 .
m1
⭈⭈⭈ Ž 1 q t n2 .
mn
/
,
which may be called the moments of . 2.3 DEFINITION. Let ⌽ Ž t 1 , . . . , t n . s w ⌽p, q Ž t 1 , . . . , t n .x1 F p F n, 1 F q F 4 be the map defined on ⺢ n by the following matrix:
⌽ Ž t1 , . . . , t n . s
1
t1
t1
t 12
Ž 1 q t12 .
Ž 1 q t12 .
Ž 1 q t12 .
Ž 1 q t12 .
.. . 1
.. . tn
Ž 1 q t n2 .
Ž 1 q t n2 .
.. . tn
Ž 1 q t n2 .
.. . t n2
.
Ž 1 q t n2 .
The function ⌽ is therefore defined on ⺢ n into Mn= 4 , where Mn=4 is the space of n = 4 matrices. 2.4 LEMMA. The function ⌽ is injecti¨ e on ⺢ n. Proof. The injectivity may be seen because of the following equality. If ⌽ Ž t 1 , . . . , t n . s w x p, q x1 F p F n, 1 F q F 4 , then we can write ti s
xi, 2 xi, 1
,
i g 1, . . . , n4 .
574
O. DEMANZE
Note that we can completely describe the range of ⌽. Specifically ⌽ Ž⺢ n . is the set of all matrices x s w x p, q x1 F p F n, 1 F q F 4 satisfying
¡x
~x
) 0, x i , 2 s x i , 3
᭙ i g 1, . . . , n4 ,
2 i, 1
q x i2, 2 q x i2, 3 q x i2, 4 s 1
᭙ i g 1, . . . , n4 ,
q
᭙ i g 1, . . . , n4 ,
x i2, 1
¢x
¦
i, 1
i, 4
x i2, 2
s xi, 1
¥.
Ž 4.
§
G0
᭙ i g 1, . . . , n4 ,
We define the linear maps B˜p, q on A by the relation A 2 f ª B˜p , q Ž f . s ⌽p , q ⭈ f g A .
Ž 5.
It follows from Ž4. that
Ý
⌽p2, q s n.
Ž 6.
1FpFn , 1FqF4
2.5 Remark. If f is in A , then f may be seen as the composition P ( ⌽ where P is a polynomial in 4 n variables. This representation is clearly not unique. Moreover, we can choose the polynomial P so that it depends only on 3n variables Ždue to the ‘‘symmetry’’ of ⌽ Ž t 1 , . . . , t n ... 2.6 THEOREM. Let be a linear positi¨ e semi-definite functional on A . Then there exist a measure , supported by ⺢ n, and a second one supported by H, included in ⺢ N with N s 4 n, such that for any polynomial P g ⺓ N w X x we ha¨ e
Ž P ( ⌽ . s
H⺢ Ž P ( ⌽ . d q HH P d ,
Ž 7.
n
n where H s Dis1 Hi with
Hi s x s w x p , q x ; x i , 1 s x i , 2 s x i , 3 s 0, x i , 4 s 1, 5 x 5 2 s n ,
½
5
i g 1, . . . , n4 . Proof. Let be as in the statement. We have Ž  < f 2 <. G 0 because for all integers i, ⌽i,2 1 q ⌽i,2 2 s ⌽i, 1 and  s ⌽ 1,11 ⭈ ⌽ 2, 21 ⭈⭈⭈ ⌽n,1n . Let H be the Hilbert space associated with the linear positive functional Žsee Remark 2.2.. We can define Ž Bp, q . on H in the same way as we have defined Ž B˜p, q . on A . The operators Bp, q become bounded
575
MOMENTS AND POSITIVITY
commuting self-adjoint operators verifying Bp2, 1 q Bp2, 2 q Bp2, 3 q Bp2, 4 s I, Bp2, 1 q Bp2, 2 s Bp , 1 , Bp , 2 s Bp , 3 , yI F Bp , q F I, 0 F Bp , q F I,
p s 1, . . . , n,
p s 1, . . . , n,
Ž 8. Ž 9.
p s 1, . . . , n,
Ž 10 .
for Ž p, q . such that 1 F p F n, 2 F q F 3, Ž 11 . for Ž p, q . such that 1 F p F n,
Ý
q g 1, 4 4 , Ž 12 .
Bp2, q s nI.
Ž 13 .
1FpFn , 1FqF4
Properties Ž8., Ž9., and Ž10. come from corresponding properties Ž4. of ⌽ Ž⺢ n .. For the remaining properties, take two elements f and g of A , N defined by f q N and g q N , and denote by f˜and ˜ g the elements of ArN respectively. Then, ² Bp , q f˜, ˜ g : s Ž ⌽p , q f ⭈ g . s Ž f ⭈ ⌽p , q g . s ² f˜, Bp , q ˜ g : . So all the operators Bp, q are symmetric. To end the proof of the properties, we may assume p equals 1 Žfor the others operators, the proof is the same because of the ‘‘symmetry’’ of the operator family.. As we have 2 2 2 2 ⌽ 1, 1 q ⌽ 1, 2 q ⌽ 1, 3 q ⌽ 1, 4 s 1,
using the linearity of and the fact that each ⌽ 1, j is real, we obtain the following equality: 5 f˜5 2 s 5 ⌽ 1, 1 f q N 5 2 q 5 ⌽ 1, 2 f q N 5 2 q 5 ⌽ 1, 3 f q N 5 2 q 5 ⌽ 1, 4 f q N 5 2 , 5 f˜5 2 s 5 B1, 1 f˜5 2 q 5 B1, 2 f˜5 2 q 5 B1, 3 f˜5 2 q 5 B1, 4 f˜5 2 . We deduce that B1, q for q in 1, . . . , 44 is a contraction, and so a bounded operator. Now we just have to show relations Ž11., Ž12., and Ž13.. As we know that ⌽ 1, 1 s ⌽ 1,2 1 q ⌽ 1,2 2 , one more time, due to the linearity of , 2 2 ² B1, 1 f˜, f˜: s Ž ⌽ 1, 1 ff . s Ž ⌽ 1, 1 ff . q Ž ⌽ 1, 2 ff .
s 5 B1, 1 f˜5 2 q 5 B1, 2 f˜5 2 G 0.
576
O. DEMANZE
Moreover, as B1, 1 is a contractive self-adjoint operator, 0 F B1, 1 F I. For B1, 4 it is exactly the same result because ⌽ 1, 4 s ⌽ 1,2 3 q ⌽ 1,2 4 . Due to the equality Ž8., we now prove that B1, 2 and B1, 3 satisfy B1,2 2 q B12, 3 F I. And so, yI F B1 , 2 s B1, 3 F I. Finally, for Ž13., we just have to use Ž6.. As all these identities and N , which is a dense subspace of H , these inequalities are true for ArN relations are also true on H. We will denote by B the self-adjoint bounded operator family Ž Bp, q .1 F p F n, 1 F q F 4 . Let Ž B . be the joint spectrum of the self-adjoint bounded operator family B. Ž B . is a subset of Ž⺢ 4 . n. So, due to the Gelfand theory, each element of the joint spectrum is either included in ⌽ Ž⺢ n . or in at least one Hi . Let x g Ž B . and let ␥ be a character associated to x. We denote by Ž␥ i, j .1 F iF n, 1 F jF 4 the family Ž␥ Ž Bi, j ..1 F iF n, 1 F jF 4 . Using properties Ž8., Ž9., Ž10., and Ž12., we verify that x s Ž␥ i, j .1 F iF n, 1 F jF 4 is in ⌽ Ž⺢ n . if each ␥ i, 1 is non-zero; if x is not in ⌽ Ž⺢ n ., there exists an element i 0 in 1, . . . , n4 so that ␥ i 0 , 1 s 0. Then by Ž9. ␥ i , 2 s ␥ i , 3 s 0 and using the relations Ž8. and Ž12. we have that 0 0 ␥ i 0 , 4 s 1. So x is necessarily in Hi 0 . Thus we have the following inclusion:
Ž B. : ⌽Ž⺢n. j H,
⌽ Ž ⺢ n . l H s ⭋.
Ž 14 .
We define now
0 s Ž B . l H 1 s Ž B . l ⌽ Ž ⺢ n . . Let E be the joint spectral measure of B; 0 and 1 are two disjoint Borel sets of ⺢ N. Then for all Borel sets of ⺢ N, we define the following positive measures:
Ž t . s ² E Ž l 0 . 1, 1: Ž . s ² E Ž l 1 . 1, 1: . We construct a third one, say , by the relation
Ž X . s Ž ⌽Ž X . . ,
᭙ X Borel sets of ⺢ n .
MOMENTS AND POSITIVITY
577
is a measure because ⌽ is bijective from ⺢ n onto ⌽ Ž⺢ n ., so we can write Ž X . s w ⌽y1 x
ž
y1
Ž X . / .
We have
Ž P ( ⌽ . s ² P Ž B . .1, 1: s
H⺢
N
P.d² E.1, 1: .
Moreover, ² E Ž . 1, 1: s ² E Ž l 0 . 1, 1: q ² E Ž l 1 . 1, 1: . Therefore,
Ž P ( ⌽ . s
H⌽ Ž⺢ .P d q H⺢ n
N
P d .
As is supported by H and by ⌽ Ž⺢ n .,
Ž P ( ⌽ . s
H⌽ Ž⺢ . P d q HH P d , n
H⌽ Ž⺢ . P d s H⺢ n
n
Ž P (⌽. d.
Consequently,
Ž P ( ⌽ . s
H⺢
n
Ž P ( ⌽ . d q H P d . H
2.7 COROLLARY. Let Ž 1 , . . . , m . be an m-tuple of real elements of A . Let Tj be polynomials in 4 n ¨ ariables such that j s Tj ( ⌽, for j in 1, . . . , m4 . Let be a linear positi¨ e semi-definite functional such that Ž j < f 2 <. G 0, ᭙ j g 1, . . . , m4 . Assume that P is in ⺓ N w X x. Then we can find two positi¨ e measures and , supported by Fj ŽTj ( ⌽ .y1 Ž⺢q . and Fj Tjy1 Ž⺢q ., respecti¨ ely, such that
Ž P ( ⌽ . s
HI Ž P ( ⌽ . d q HH P d , X
where I s Fj ŽTj ( .y1 Ž⺢q . and H X s H l Fj Tjy1 Ž⺢q ..
Ž 15 .
578
O. DEMANZE
Proof. As Ž j < f 2 <. is positive for all integers j g 1, . . . , m4 , we obtain the condition Tj Ž B . positive for all integers j because ²Tj Ž B . f˜, f˜: s ²Tj ( ⌽ f˜, f˜: s ² j f˜, f˜: s j < f < 2 G 0.
ž
/
Let ␥ be a character; hence Tj Ž␥ Ž B .. s ␥ ŽTj Ž B .. G 0. We deduce that ␥ Ž B . g Tjy1 Ž⺢q .. Consequently, the spectral measure associated with the joint spectrum Ž B . has its support included in Fj Tjy1 Ž⺢q .. 3. POLYNOMIAL REPRESENTATIONS In this section, we shall use the representation of a linear positive semi-definite functional on A into the sum of two integrals with respect to two positive measures to describe the structure of some positive polynomials. 3.1 LEMMA. Let S be a real linear space, and let C be a positi¨ e con¨ ex cone included in S with the property S s C y C . Assume that C can be written as Dd G 1 Cd where the Cd is a con¨ ex cone such that the Sd s Cd y Cd are linear spaces of finite dimension, and such that Cdq 1 l Sd s Cd for all strictly positi¨ e integers d. Denote by intŽ Cd . the relati¨ e interior of Cd as a subset of the Euclidean space Sd . Assume that there exists an element g C1 with the property that, for all strictly positi¨ e d and for all linear non-zero positi¨ e functionals in SdU on Cd , one has Ž . ) 0. Let r 0 g Sd 0 y intŽ Cd 0 ., where d 0 is a strictly positi¨ e integer; then there exists a linear functional defined on S into ⺢ such that Ž r 0 . F 0 and is strictly positi¨ e on intŽ Cd . for all d more than d 0 . In particular restricted to C is positi¨ e. We can find the proof of the above lemma in w7x Žwhich, in turn, uses a method due to Cassier; see w3x.. 3.2 DEFINITION.
For all positive integers d, we define
Fd s g Ž t . s
½
Ý a␣ ,  t ␣ Ž t . 
␣, 
with ␣ F 2  F 4 D ,
5
D s Ž d, d, . . . , d . . If P is a polynomial defined on ⺢ n, we denote by ␦ j Ž P . the degree of P as a one variable polynomial in X j on the Euclidean ring ⺢w X 1 , . . . , X jy1 ,
579
MOMENTS AND POSITIVITY
X jq1 , . . . , X n x. Then, we denote by ␦ Ž P . the ‘‘multi-degree’’ of P, i.e., the following multi-index:
␦ Ž P . s ␦ 1 Ž P . , . . . , ␦n Ž P . 4 .
Ž 16 .
Let C be the positive cone generated by the elements of the form r 2 , r g A and Pj ␥ Ž P j . s 2 , s g A , where P1 , . . . , Pk are fixed real polynomials. In the same way, let ⌺ be the positive cone included in C , generated by the elements of the form r 2 , r g A . Let P Ž X . s Ý ␣ g ␣ X ␣ be a polynomial in n variables of even multi-degree. We can associate P with an other polynomial P˜ in 3n variables via the formula P˜Ž X , Y , Z . s
Ý g␣ X  Y ␣y2  Z ␦ Ž P .r2q y ␣ ,
Ž 17 .
␣
or equivalently, P˜Ž X , Y , Z . s Z ␦ Ž P .r2
Ý g␣ ␣
X

Y
␣ y2 
ž /ž / Z
Z
,
Ž 18 .
where  is the multi-index Žw ␣ 1r2x, . . . , w ␣ nr2x. Žw.x is the integer part function.. This polynomial P˜ will be composed on the right with the function ⌽ Žregarding P˜ as a 4 n variable polynomial.. To understand this representation, it is easier to index again the different variables. Recalling that every element of ⌽ Ž⺢ n . can be expressed by a matrix x s w x p, q x1 F p F n, 1 F q F 4 , let us write P˜ in the following form: P˜Ž X 1, 1 , . . . , X n , 4 . s
Ý g␣ X1, 4 1
␣
⭈⭈⭈ X n,n4 X 1␣, 12y2  1 ⭈⭈⭈
X n␣,n2y2  n X 1␦ 1Ž P .r2q  1y␣ 1 ⭈⭈⭈ X n␦,n1Ž P .r2q  ny␣ n . Ž 19 . This means that the n variables X, Y, and Z play the role of Ž ⌽ 1, 4Ž t ., . . . , ⌽n, 4 Ž t .., Ž ⌽ 1, 2 Ž t ., . . . , ⌽n, 2 Ž t .., and Ž ⌽ 1, 1Ž t ., . . . , ⌽n, 1Ž t .., respectively. An easy calculation gives the relations P ␦ Ž P .r2 Ž t . s P˜( ⌽ Ž t . , P˜Ž tX , t , 1 . s P Ž t . ,
᭙t g ⺢ n. ᭙t g ⺢n,
Ž 20 . Ž 21 .
where tX s Ž t 12 , . . . , t n2 .. Formula Ž19. gives an explicit way to find the polynomial P˜ associated with the polynomial P.
580
O. DEMANZE
3.3 THEOREM. Let Ž P1 , . . . , Pk . be a k-tuple of polynomials of multi-degrees 2 D 1 , . . . , 2 D k , respecti¨ ely, and let P be an other polynomial, strictly k positi¨ e on Fis1 Piy1 Ž⺢q ._ 04 , of multi-degree 2 D. Let P˜1 , . . . , P˜k , P˜ be their associated polynomials. Assume that P˜Ž t . ) 0 for each t in H l k Fis1 P˜iy1 Ž⺢q . where H has been defined in Theorem 2.6. Then there exist a positi¨ e multi-index M and a finite number of real polynomials ql 4l g L and qi, l 4i g 1, . . . , k4, l g LX such that PŽ t. s Ž t.
2M
k
ž
Ý ql2 Ž t . q Ý Ý Pi Ž t . qi2, l Ž t . X
lgL
is1 lgL
t g ⺢n,
/
,
n M g ⺪q .
Ž 22 .
Proof. We want to apply Lemma 3.1, so we begin to verify the conditions of this lemma. Set Cd s C l Fd and Sd s Cd y Cd , which is a linear space. As Cd ; Sd ; Fd , Sd is a finite dimensional space. Moreover, A s C y C , because every element f in S can be written 14 Žw f q 1x 2 y w f y 1x 2 .. We have
Ž Cdq 1 l Sd . ; Ž Cdq1 l Fd . s Ž C l Fdq1 l Fd . s Ž C l Fd . s Cd . Due to the three following inclusions Ž Cdq 1 l Sd . ; Cd , Cd ; Cdq1 , Cd ; Sd , we have Ž Cdq1 l Sd . s Cd . We must find now an element g C1 such that, for all strictly positive d and for every non-zero linear functional in SdU which is positive on Cd , we have Ž . ) 0. Choose s 1, fix an integer d, let in SdU , assume that Ž . s 0, and let us prove that is zero in Sd . It suffices to verify that s 0 on C . Let F 2 D Ž t . s Ž1 q t 12 . 2 d ⭈⭈⭈ Ž1 q t n2 . 2 d ; then F 2 D Ž t . is a sum of squares where t 2 ␣ s t 12 ␣ 1 ⭈⭈⭈ t n2 ␣ n appears with ␣ i less than 2 d for every i s 1, . . . , n. So, 1y
t2␣ F2DŽ t.
g Cd ,
᭙␣ F 2 D.
As Ž1. s 0, then y w t 2 ␣rF 2 D Ž t .x G 0, but w t 2 ␣rF 2 D Ž t .x s wŽ t ␣r F D Ž t .. 2 x G 0. Thus,
t2␣ F2DŽ t.
s 0,
᭙␣ F 2 D.
Ž 23 .
581
MOMENTS AND POSITIVITY
Let ␣ F 2 D and  F 2 D and let g ⺢; then t ␣rF D Ž t . and t rF D Ž t . are in Fd , so
ž
t␣
q
F DŽ t. s 2
2
t F DŽ t.
t2 F2DŽ t.
/
q 2
t ␣q  F2DŽ t.
t2␣
q
F2DŽ t.
G 0.
According to Ž23., t2
F2DŽ t.
t2␣
s
s 0.
F2DŽ t.
Since for every real number , w t ␣q rF 2 D Ž t .x is positive, one has
t ␣q 
᭙␣ F 2 D,
s 0,
F2DŽ t.
᭙ F 2 D.
Ž 24 .
If ␣ F 4 D, write ␣ s ␣ 1 q ␣ 2 with ␣ 1 F 2 D and ␣ 2 F 2 D. Applying the above result with ␣ 1 and ␣ 2 we obtain
t ␣ 1q ␣ 2
s
F2DŽ t.
t␣ F2DŽ t.
᭙␣ F 4 D.
s 0,
Ž 25 .
We have just to see that w t ␣rF  Ž t .x s 0 when ␣ F 2  F 4 D. Let i g 1, . . . , n4 , and let i s 2 k i q i , with i g 0, 14 . Then t␣ F Ž t. t␣ FŽ t.
t i␣ i
n
s
Ł
is1
Fi2 k iq i Ž t . t i␣ i
n
s
Ł
is1, is0
where Fi Ž t . s 1 q t i2 ,
,
t i␣ i
n
Fi2 k iq i
.
Ł
is1, is1
Fi2 k iq i Ž t .
.
Let I s i such that i s 04 and I X s i such that i s 14 . Also let Ž e1 , . . . , e n . be the canonical basis of ⺢ n. We have two cases. Ža. If i g I, then ␣ F 2  F 4 D becomes ␣ i F 4 k i F 4 d, t i␣ i F Ž2 k i . e i Ž t .
s
t i␣ i FiŽ2 dy2 k i . e i Ž t . F Ž2 d. e i Ž t .
,
582
O. DEMANZE
with the degree of the numerator equal to ␣ i q 2Ž2 d y 2 k i ., which is less than 4 d. Žb. If i g I X , then ␣ F 2  F 4 D becomes ␣ i F 4 k i q 2 F 4 d, t i␣ i F Ž2 k iq1. e i Ž t .
s
t i␣ i F e i Ž t . F Ž2 k iq2. e i Ž t .
.
As 2 k i q 1 F 2 d, 2 k i q 2 F 2 d, we have t i␣ i F Ž2 k iq1. e i Ž t .
s
t i␣ i F e i F Ž2 dy2 k iy2 . e i Ž t . F Ž2 d. e i Ž t .
,
with the degree of the numerator equal to ␣ i q 2 q 2.Ž2 d y 2 k i y 2., which is less than 4 d too. Writing that t␣ FŽ t.
s
t ␣ F 2 Dy  Ž t . F2DŽ t.
,
when ␣ F 2  F 4 D, we can now use Ž25.. We conclude that
t␣ FŽ t.
s0
when ␣ F 2  F 4 D.
Ž 26 .
So s 0 in Fd , and we can apply Lemma 3.1. Clearly, P ␦ Ž P .r2 g Fd s Sd . Assume that P ␦ Ž P .r2 f IntŽ Cd .; then there exists a linear functional ⌿ in A into ⺓ such that its restriction to C is positive and ⌿ Ž P ␦ Ž P .r2 . F 0. Because of the positivity of ⌿ on C , we can apply Corollary 2.7. So there exist two positive measures and such that ⌿ Ž P ␦ Ž P .r2 . s
HI P˜( ⌽ d q HH P˜d , X
where P˜( ⌽ s P ␦ Ž P .r2 , I s Fj Pjy1 Ž⺢q ., and H X s H l Fj P˜jy1 Ž⺢q .. Assuming that has an empty support, then the spectrum Ž B . is in the range by ⌽ of the support suppŽ . of , which is an nonempty set. But we have assumed that P was strictly positive in I, so it is the same for P ␦ Ž P .r2 which is P˜( ⌽. In this case HI P˜( ⌽ d ) 0. If has a nonempty support, as we have assumed that P˜ ) 0 in suppŽ ., HH X P˜d ) 0. The two
583
MOMENTS AND POSITIVITY
measures cannot have simultaneously an empty support because Ž B . is a nonempty set, and so in every case 0 G ⌿ Ž P ␦ Ž P .r2 . s
H⺢
n
P˜( ⌽ d q
HH P˜d ) 0.
But this is impossible, so P ␦ Ž P .r2 is in the interior of Cd , and in particular in Cd . Hence there exist a finite number of elements in A ql 4l g L and qi, l 4i g 1, . . . , k4, l g LX such that PŽ t. Ž t.
␦ Ž P .r2
k
s
Ž . Ý ql2 Ž t . q Ý Ý Pi Ž t . Ž t . ␦ P qi2, l Ž t . , i
X
lgL
t g ⺢n;
is1 lgL
that is, k
PŽ t. s
X
Ý
l gL 1
q1,2 lX Ž t . q
Ý Ý is1 l
X
X gL 1
Pi Ž t . Ž t .
␦ Ž Pi .
q1,2 i , lX Ž t . ,
t g ⺢n,
where q1, lX 4lX g L 1 and q1, i, lX 4i g 1, . . . , k4, lX g LX1 are not necessarily in A , but n. their denominators are of the form w Ž t . F xy1 Ž F g ⺪q . Taking the same denominator, because ␦ Ž Pi . s 2 Di for all integers i, we derive PŽ t. s Ž t.
2M
k
ž
Ý
X
X
l gL
r l2X Ž t . q
Ý Ý Pi Ž t . ri2, l Ž t . X
X
X
is1 l gL
/
,
t g ⺢n,
where r lX , ri, lX 4lX g LX , i g 1, . . . , k4 are polynomials in ⺢ nw X x. 3.4 COROLLARY. If P is a polynomial in n ¨ ariables such that P˜ is strictly n positi¨ e in ⺢ N _ 04 , then there exists M in ⺪q such that PŽ t. s Ž t.
2M
t g ⺢n,
2 l
½ Ý q Ž t. 5 , lgL
where ql 4l g L are polynomials in ⺢ nw X x. Proof. As P˜ is a strictly positive polynomial on ⺢ N _ 04 , P is strictly positive on ⺢ n because of Ž21.. So the multi-degree of P is even. We can apply the above result with all polynomials Pj equal to 0. 3.5 THEOREM.
Let P be a strictly positi¨ e polynomial on ⺢ n such that PŽ t. s
ž
Ý
g 0, 1 4 n _ 0 4
b M t 2 M q Q Ž t . ,
/
584
O. DEMANZE
where b M ) 0 for all . If 2 M s Ž2 m1 , . . . , 2 m n . is the multi-degree of P, n assume that ␦ Ž Q . - 2 M. Then there exists N in ⺪q such that PŽ t. s 2NŽ t.
2 l
½ Ý q Ž t. 5 ,
t g ⺢n,
lgL
where Ž ql . l g L are polynomials in ⺢ nw X x. Proof. Let ␦ Ž P . be the multi-degree of P, ␦ Ž P . s Ž2 m1 , . . . , 2 m n .. Then P Ž t . ␦ Ž P .r2 Ž t . is in A . Let ⌺ be defined as in the beginning of the section. As in Theorem 3.3, one can use Lemma 3.1 due to w7x, letting ⌺ d s Fd l ⌺. Assume that P Ž t . ␦ Ž P .r2 Ž t . f IntŽ ⌺ d .; then there exists a linear functional ⌿ on A into ⺓ such that the restriction of ⌿ to ⌺ is positive and such that ⌿ Ž P ␦ Ž P .r2 . F 0. Because of the positivity of ⌿ in ⌺, Theorem 2.6 can be applied. Hence there exist two positive measures and such that ⌿ Ž P ␦ Ž P .r2 . s
H⺢
n
s
H⺢
n
P˜( ⌽ d q
HH P˜d
P Ž t . ␦ Ž P .r2 Ž t . d Ž t . q
HH P˜d .
If suppŽ . / ⭋, then H⺢ n P Ž t . ␦ Ž P .r2 Ž t . d Ž t . ) 0. If suppŽ . s ⭋, then suppŽ . / ⭋, and let g suppŽ ., g Ž⺢ 4 . n. Then there exists an integer j0 in 1, . . . , n4 such that
Ž j , 1 , j , 2 , j , 3 , j , 4 . s Ž 0, 0, 0, 1. 0
0
0
0
Ždecomposition of Ž B . into two disjoint parts included respectively in ⌽ Ž⺢ n . and in H .. Via a change of indices, we may assume j0 s 1. Let M X be the multi-index Ž m 2 , . . . , m n .. We have P Ž t . ␦ Ž P .r2 Ž t . s
Ý
g 0, 1 _ 0 4 4n
b M t 2 M ␦ Ž P .r2 Ž t .
q ␦ Ž P .r2 Ž t . Q Ž t . . s
Ý
Xg 0, 1 4 ny1 _ 0 4
q
Ý
Xg 0, 1 4 ny1
b 0, X M X t Ž0, 2
X
X
M . ␦ Ž P .r2
bm 1 , X M X t 12 m 1 t Ž0, 2
q ␦ Ž P .r2 Ž t . Q Ž t . .
X
X
Ž t.
M . ␦ Ž P .r2
Ž t.
585
MOMENTS AND POSITIVITY
So, using formula Ž19., we obtain P˜Ž x . s
Ý
Xg 0, 1 4 ny1 _ 0 4
q
Ý
Xg 0, 1 4 ny1
b 0, X M X Ž x 1, 1 .
bm 1 , X M X Ž x 1, 4 .
m1
Ž ˆx 1 .
m1
XM X
Ž ˆx 1 .
XM X
q x 1, 1 Q1 Ž x . q x 1, 2 Q2 Ž x . , where Ž ˆ x1.
X
M
X
j.m j s Ł njs2 x j, j4m j x Ž1y . As Ž 1, 1 , 1, 2 , 1, 3 , 1, 4 . s Ž0, 0, 0, 1., j, 1
P˜Ž . s
Ý
Xg 0, 1 4 ny1 X
bm 1 , X M X Ž ˆ1 .
XM X
,
X
j.m j where Ž ˆ1 . M s Ł njs2 j,4j m j j,Ž1y . 1 X Let b s minw bm 1 , X M X , g 0, 14 ny 1 x. Because every j, 4 and every j, 1 is positive
P˜Ž . G b
Ý
Xg 0, 1 4 ny1
Ž ˆ1 .
XM X
n
sb X
.m . Ý j,4m j,Ž1y 1 j
Ý
j
j
j
g 0, 1 4 ny1 js2 j.m j As in the sum Ý X g 0, 14ny 1 Ł njs2 j,4j m j j,Ž1y every element of the form 1 mj n Ł js2 j, *j appears, where *j take the values 1 and 4, we obtain the following equality:
n
X
Ý
j.m j s Ł j,4j m j j,Ž1y 1
g 0, 1 4 ny1 js2
n
Ł Ž j,m1 q j,m4 . . j
j
js2
Therefore, n
P˜Ž . G b Ł Ž j,m1 j q j,m4 j . . js2
Since j, 1 and j, 4 are positive and cannot be zero simultaneously, for all j Žas is in the spectrum of B, using Ž8., Ž9., and Ž12., we know that if j, 1 s 0 then j, 4 s 1., one obtains that P˜Ž . ) 0; thus HH P˜d ) 0. The two supports cannot be empty simultaneously, so 0 G ⌿ Ž P ␦ Ž P .r2 . s
H⺢
n
P˜( ⌽ d q
HH P˜d ) 0,
586
O. DEMANZE
where P˜( ⌽ s P ␦ Ž P .r2 . That is impossible; thus P ␦ Ž P .r2 is in the interior of one ⌺ d and, in particular, in ⌺ d . So there exists a finite number of elements of A , ql 4l g L , such that PŽ t. Ž t.
␦ Ž P .r2
s
Ý ql2 Ž t . ,
t g ⺢ n.
lgL
Consequently, there exists a finite number of elements of ⺢ nw X x, qXlX 4lX g LX , such that PŽ t. s 2NŽ t.
Ý
X
X
l gL
qXlX2 Ž t . ,
t g ⺢ n.
Theorem 3.5 gives a representation of strictly positive polynomials on ⺢ n, where we do not assume any condition on the associated polynomial ˜ This result can be seen as a result similar to the Artin theorem which P. gives an answer to the 17th Hilbert problem ŽParis 1900.; see w2, 6x. 3.6 EXAMPLE. It is known that the polynomial x 12 x 22 Ž x 12 q x 22 y 1. q 1 is strictly positive on ⺢ 2 Žbut it cannot be written as a sum of squares of polynomials; see w1x.. Let P Ž x 1 , x 2 . s x 16 x 26 q x 16 q x 26 q x 12 x 22 Ž x 12 q x 22 y 1 . q 1. An easy calculation gives P˜Ž X , Y , Z . s X 13 X 23 q X 13 Z23 q Z13 X 23 q X 12 Z1 X 2 Z22 q X 1 Z12 X 22 Z2 y X 1 Z12 X 2 Z22 q Z13 Z23 . In particular, P˜Ž X, Y, 0. s X 13 X 23, so P˜ does not satisfy the conditions of Corollary 3.4. Nevertheless, using Theorem 3.5, P is in the positive cone ⌺; i.e., it can be written as a sum of squares of fractions where the 2 denominators of these fractions are of the form of w F xy1 with F in ⺪q . REFERENCES 1. C. Berg, J. P. R. Christensen, and P. Ressel, ‘‘Harmonic Analysis on Semigroups,’’ Springer-Verlag, New YorkrBerlinrHeidelbergrTokyo, 1984. 2. J. Bochnak, M. Coste, and M. M. Roy, ‘‘Geometrie algebrique reelle,’’ Springer-Verlag, ´ ´ ` ´ BerlinrHeidelbergrNew YorkrLondonrParisrTokyo, 1987. 3. G. Cassier, Probleme de polynomes ` des moments sur un compact de ⺢ n et decomposition ´ ˆ `a plusieurs variables, J. Funct. Anal. 58 Ž1984., 254᎐266. 4. N. Dunford and J. T. Schwartz, ‘‘Linear Operators, Part II,’’ Interscience, New Yorkr LondonrSydney, 1963.
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5. B. Fuglede, The multidimensional moment problem, Exposition. Math. 1 Ž1983., 47᎐65. 6. N. Jacobson, ‘‘Theory of Fields and Galois Theory,’’ Lectures in Abstract Algebra III, Van Nostrand, TorontorNew YorkrLondon, 1975. 7. M. Putinar and F. H. Vasilescu, Solving moment problems by dimensional extension, Ann. Math. 149 Ž1999., 1087᎐1107. 8. I. R. Shafarevich, ‘‘Basic Algebraic Geometry. 1,’’ Springer-Verlag, Berlin, 1994.