Accessibility / Report Error

Positive Polynomials on Closed Boxes

M. A. DINIZ R. B. STERN L. E. SALASAR About the authors

ABSTRACT

We present two different proofs that positive polynomials on closed boxes of 2 can be written as bivariate Bernstein polynomials with strictly positive coefficients. Both strategies can be extended to prove the analogous result for polynomials that are positive on closed boxes of n, n>2.

Key words
polinômios positivos; hipercubo unitário; polinômios de Bernstein

RESUMO

Apresentamos duas demonstrações, por métodos diferentes, de que polinômios positivos em caixas fechadas no 2 podem ser escritos como polinômios de Bernstein bivariados com coeficientes estritamente positivos. Ambas as estratégias de demonstração podem ser estendidas para provar o resultados análogo para polinômios que são positivos em caixas fechadas no n, n>2.

Palavras-chave
positive polynomials,; unit box; Bernstein polynomials

1 INTRODUCTION

The goal of this paper is to show that real polynomials that are strictly positive on closed boxes have a representation with positive coefficients when written using Bernstein’s polynomial basis. More specifically, we will prove the result for the unit box I=[0,1]×[0,1], i. e. we present new proofs for the following theorem:

Theorem 1.1. If p:2 is such that

p(x1,x2)=i=0n1j=0n2ai,jx1ix2j(1.1)
and, for every (x1,x2)I, p(x1,x2)>0, then there exist q1n1,q2n2 and Ci,j>0, (i,j)Q1×Q2, such that
p(x1,x2)=i=0q1j=0q2Ci,jx1i(1x1)q1ix2j(1x2)q2j,
where Q1={0,1,,q1} and Q2={0,1,,q2}.

Furthermore, we constructively derive the values of q1 and q2.

Theorem 1.1 is an extension of similar results obtained for positive polynomials on compact intervals and multidimensional simplexes by, respectively, Bernstein [1][1] S.N. Bernstein. Sur la représentation des polynômes positifs. Communications de la Société mathématique de Kharkow, 14(5) (1915), 227–228., Hausdorff [4][4] F. Hausdorff. Summationsmethoden und momentfolgen. I. Mathematische Zeitschrift, 9(1-2) (1921), 74–109. and Pólya [6][6] G. Pólya. Über positive darstellung von polynomen. Vierteljschr. Naturforsch. Ges. Zürich, 73 (1928), 141–145.. We are aware that, using a different proof strategy, Cassier [2][2] G. Cassier. Problème des moments sur un compact de rn et représentation de polynômes à plusieurs variables. Journal of Functional Analysis, 58 (1984), 256–266. has proven a general result from which a similar version of Theorem 1.1 follows. We discuss this more extensively at the final section.

We provide two proofs of Theorem 1.1. The first one is supported by results for the univariate version of Theorem 1.1, proved by Powers and Reznick [7][7] V. Powers & B. Reznick. Polynomials that are positive on an interval. Transactions of the American Mathematical Society, 352(10) (2000), 4677–4692.. The second proof extends the approach in Garloff [3][3] J. Garloff. Convergent bounds for the range of multivariate polynomials. In “International Symposium on Interval Mathematics”. Springer (1985), pp. 37–56. and Rivlin [8][8] T. Rivlin. Bounds on a polynomial. Journal of Research of the National Bureau of Standards, 74 (1970), 47–57..

The paper is organized as follows. Section 2 establishes notation and brings the relevant definitions used in the paper. In Section 3 we present the auxiliary results. These results are used in one of the proofs of Theorem 1.1, given in Section 4. Section 5 brings an alternative proof, based on [3][3] J. Garloff. Convergent bounds for the range of multivariate polynomials. In “International Symposium on Interval Mathematics”. Springer (1985), pp. 37–56. and [8][8] T. Rivlin. Bounds on a polynomial. Journal of Research of the National Bureau of Standards, 74 (1970), 47–57..

2 DEFINITIONS AND NOTATION

Definition 1. Let 𝒫n be the linear space of polynomials of degree n, i.e.

𝒫n={p:, where ai,0in:p(x)=i=0naixi}.

Definition 2. For any p𝒫n we define its Goursat transform p̃ by

p ̃ ( x ) = ( 2 x ) n p ( 1 x x ) .

Definition 3. Let n+ be the set of polynomials of degree n that can be written with non-negative coordinates in the Bernstein basis,

n + = { p 𝒫 n , where A i 0 : p ( x ) = i = 0 n A i x i ( 1 x ) n i } .

Similarly, let n+,* be the set of polynomials of degree n that can be written with positive coordinates in the Bernstein basis,

n+,*={p𝒫n, where Ai>0:p(x)=i=0nAixi(1x)ni}.

Definition 4. For every a=(a1,,an)n, mn and 0im, let

A i , m ( a ) = j = 0 min ( n , i ) ( m j m i ) a j .

Definition 5. For every a=(a1,,an)n, let

B k ( a ) = ( i , j ) 2 : i j = n k ( 2 n ( 1 ) j ( i j ) a i ) .
Notice that Bk(a) is a linear combination of a.

Definition 6. For each 0in1 and 0jn2, let ai,j. For each 0in1, define ai: as

a i ( x 2 ) = j = 0 n 2 a i , j x 2 j .
Also define a(x2)=(a0(x2),,an1(x2)).

Definition 7. For each 0in1 and 0jn2, let ai,j. For each mn1 and 0kn2, define

b k , i , m ( a ) = j = 0 min ( n 1 , i ) ( m j m i ) a j , k .
Also define bi,m(a)=(b0,i,m(a),,bn2,i,m(a)).

3 AUXILIARY RESULTS

Lemma 3.1. If p𝒫n, p(x)=i=0naixi, then, for every mn,

p ( x ) = i = 0 m A i x i ( 1 x ) m i
if and only if
A i = A i , m ( a ) , a = ( a 1 , , a n ) . (3.1)

Proof. Applying the Binomial Theorem to the identity xi=xi(1x+x)mi, it follows that

xi=j=im(miji)xj(1x)mj.
From this expression, we obtain that
p(x)=i=0mAi,m(a)xi(1x)mi.
The proof that the Ai’s are unique follows from observing that, {xi(1x)mi:0im} is a basis for 𝒫m.

The following theorem is a consequence of Theorem 6 in [7][7] V. Powers & B. Reznick. Polynomials that are positive on an interval. Transactions of the American Mathematical Society, 352(10) (2000), 4677–4692..

Theorem 3.2. Let p𝒫n be such that p(x)>0 for all x[0,1]. Let λ=minx[0,1]p(x) and ej be such that p̃(x)=i=0nejxj. If q3n+2n2maxj|ej|λ+1, then pq+.

Lemma 3.3. Let p𝒫n be such that p(x)>0 for all x[0,1]. Let λ=minx[0,1]p(x) and ej be such that p̃(x)=i=0nejxj. Let q=3n+2n2maxj|ej|λ+1, where y=min{n:ny}. Then, for every q*2q, pq*+,*.

Proof. It follows from Theorem 3.2 that there exist Ai0 such that

p(x)=i=0qAixi(1x)qi.
Note that
p(x)=i=0qAixi(1x)qi=i=0qAixi(1x)qi(x+1x)q*q=i=0qAixi(1x)qij=0q*q(q*qj)xj(1x)q*qj=k=0q*(l=max(0,k+qq*)min(q,k)(q*qkl)Al)xk(1x)q*k.

Observe that, for every k, l=max(0,k+qq*)min(q,k)(q*qkl)Almin(A0,Aq)>0, since A0=p(0)>0 and Aq=p(1)>0. Therefore, pq*+,*.

Lemma 3.4. If p(x)=i=0naixi and a=(a1,,an), then

p ̃ ( x ) = k = 0 n B k ( a ) x k .

Proof.

p ̃ ( x ) = ( 2 x ) n p ( 1 x x ) = ( 2 x ) n i = 0 n a i ( 1 x x ) i = i = 0 n 2 n a i ( 1 x ) i x n i = i = 0 n 2 n a i j = 0 i ( i j ) ( 1 ) j x n i + j = k = 0 n ( i , j ) : i j = n k ( 2 n ( 1 ) j ( i j ) a i ) x k = k = 0 n B k ( a ) x k .

4 PROOF OF THEOREM 1.1

The main idea behind this proof is to use twice the positive representation result for univariate polynomials (Lemma 3.3). For every fixed value in one of the coordinates of a bivariate polynomial, the function of the free coordinate is a univariate polynomial. This polynomial admits a positive Bernstein representation. Furthermore, the coefficients of this representation are univariate polynomials on the coordinate that was fixed, allowing another application of the positive Bernstein representation theorem for univariate polynomials. As a result of both applications, a positive Bernstein representation for the bivariate polynomial is obtained. This strategy can be extended by induction to arbitrary n-variate polynomials.

Proof. For a given x2[0,1], obtain from definition 6 that

px2(x1)=p(x1,x2)=i=0n1ai(x2)x1i,
Thus, px2𝒫n1 and px2(x1)>0 for all x1[0,1]. From this observation, one can obtain two facts. First, since I is compact, then λ=inf(x1,x2)I2p(x1,x2)>0 and
λx2=infx1[0,1]px2(x1)λ>0.(4.1)

Second, it follows from Lemma 3.4 that

p̃x2(x1):=i=0n1Bi(a(x2))x1i.
Since each Bi is a linear combination of the elements of a(x2) and each element of a(x2) is a polynomial on x2, Bi(a(x2)) is a polynomial on x2. Since [0,1] is compact, there exists L< such that
supx2[0,1]maxi|Bi(a(x2))|=L.(4.2)
Therefore, it follows from Lemma 3.3 and Equations (4.1) and (4.2) that, taking q1=2(3n1+2n12supx2[0,1]maxi|Bi(a(x2))|inf(x1,x2)I2p(x1,x2)+1), one obtains that, for all x2[0,1], px2q1+,*. Therefore, it follows from Lemma 3.1 that, for all x2[0,1],
p(x1,x2)=px2(x1)=i=0q1Ai,q1(a(x2))x1i(1x1)q1i(4.3)
where Ai,q1(a(x2))>0. Notice that
Ai,q1(a(x2))=j=0min(n1,i)(q1jq1i)aj(x2)=j=0min(n1,i)(q1jq1i)k=0n2aj,kx2k=k=0n2(j=0min(n1,i)(q1jq1i)aj,k)x2k=k=0n2bk,i,q1(a)x2k𝒫n2

It follows from Lemma 3.3 that, taking q2=2(3n2+maxi2n22maxj|Bj(bi,q1(a))|infx2IAi,q1(a(x2))+1), one obtains that

Ai,q1(a(x2))=j=0q2Ci,jx2j(1x2)q2j, 0iq1(4.4)
where Ci,j>0. By applying Equation (4.4) to Equation (4.3), one obtains
p(x1,x2)=i=0q1j=0q2Ci,jx1i(1x1)q1ix2j(1x2)q2j.

5 ALTERNATIVE PROOF

We consider, as before, the bivariate polynomial p given in (1.1) and λ=inf(x1,x2)Ip(x1,x2). For q1,q21, let us define the bivariate polynomial

bk,l(q1,q2)(x1,x2)=(q1k)x1k(1x1)q1k(q2l)x2l(1x2)q2l,(5.1)
where kQ1 and lQ2. The set of polynomials {bk,l(q1,q2)(x1,x2),kQ1,lQ2} are the Bernstein polynomials of degree q1 and q2 and form a basis for the linear space of all bivariate polynomials of the form (1.1) with n1=q1 and n2=q2.

Lemma 5.1. If iQ1 and jQ2, then

x 1 i x 2 j = k = 0 q 1 l = 0 q 2 ( k i ) ( l j ) ( q 1 i ) ( q 2 j ) b k , l ( q 1 , q 2 ) ( x 1 , x 2 ) , (5.2)
where it is assumed that (mv)=0 for integers m and v such that m<v.

Proof. The result follows by applying the Binomial Theorem to the identity x1ix2j=x1i(1x1+x1)q1ix2j(1x2+x2)q2j.

Henceforth, we shall consider q1n1,q2n2. Then, it follows from Lemma 5.1 that p(x1,x2) given in (1.1) can be rewritten as

p(x1,x2)=k=0q1l=0q2ck,lq1,q2bk,l(q1,q2)(x1,x2),(5.3)
where
ck,lq1,q2=i=0n1j=0n2ai,j(ki)(lj)(q1i)(q2j).(5.4)
The ck,l(q1,q2) are the Bernstein coefficients and (5.3) is the Bernstein form of p(x1,x2). In the sequel, we denote by
c(q1,q2)=min(k,l)Q1×Q2ck,l(q1,q2)
the smallest Bernstein coefficient of p(x1,x2).

Theorem 5.2. If p is given by (1.1), then

λ c ( q 1 , q 2 ) 0 . (5.5)

Proof. Since bk,l(q1,q2)(x1,x2)0 for all (x1,x2)I, then

c(q1,q2)=k=0q1l=0q2c(q1,q2)bk,l(q1,q2)(x1,x2)k=0q1l=0q2ck,l(q1,q2)bk,l(q1,q2)(x1,x2)=p(x1,x2),
for all (x1,x2)I, which implies the assertion.

Theorem 5.3. If p is given by (1.1), q1n1 and q2n2, then

λ c ( q 1 , q 2 ) γ 1 ( q 1 1 ) q 1 2 + γ 2 ( q 2 1 ) q 2 2 ,
where
γ 1 = 1 2 i = 0 n 1 j = 0 n 2 | a i , j | i ( i 1 ) , γ 2 = 1 2 i = 0 n 1 j = 0 n 2 | a i , j | j ( j 1 ) .

Proof. For any real function f(x1,x2), define its Bernstein approximation on I by

Bq1,q2(f;x1,x2)=k=0q1l=0q2f(kq1,lq2)bk,l(q1,q2)(x1,x2).(5.6)

For 0in1 and 0jn2, let δk,lq1,q2(i,j), (k,l)Q1×Q2, be the Bernstein coefficients of the polynomial Bq1,q2(x1ix2j;x1,x2)x1ix2j, i.e.,

Bq1,q2(x1ix2j;x1,x2)x1ix2j=k=0q1l=0q2δk,lq1,q2(i,j)bk,l(q1,q2)(x1,x2).(5.7)

Then, from Lemma 5.1 and (5.6) , it follows that

δk,lq1,q2(i,j)=(kq1)i(lq2)j(ki)(lj)(q1i)(q2j),(5.8)
kQ1,lQ2.

For any fixed 0in1 and 0jn2, we can prove that

0δk,lq1,q2(i,j)(q11q12)i(i1)2+(q21q2)j(j1)2,(5.9)
for all kQ1 and lQ2. In order to prove (5.9), it suffices to show that
0φkq1(i)=(kq1)i(ki)(q1i)(q11q12)i(i1)2,for all kQ1,(5.10)
0φlq2(j)=(lq2)j(lj)(q2j)(q21q22)j(j1)2,for all lQ2.(5.11)

Since (5.11) is essentially the same as (5.10), we only present the proof of (5.10). Notice that (5.10) clearly holds for i=0, i=1, k=0 and k=q1. Thus, let us consider 1kq11 and i2.

If k<i, then

0φkq1(i)=(kq1)i(kq1)2(q11q1)(i1q1)(q11q12)i(i1)2.

If ki, then

φkq1(i)=(kq1)i(ki)(q1i)=(kq1)i[1r=0i1(1r/k)(1r/q1)].

Since 0(1r/k)(1r/q1)1 for all r=0,,i1, it follows that

0φkq1(i)(kq1)i[1r=0i1(1rk)].(5.12)

Using the fact that, for any z1,,zm[0,1], we have

i=1m(1zi)1i=1mzi,
it follows from (5.12) that
0φkq1(i)(kq1)ii(i1)2k=(kq1)i1i(i1)2q1(q11q12)i(i1)2,(5.13)
which finishes the proof of (5.10) and consequently proves (5.9).

Considering the form (1.1) of p(x1,x2) and the Bernstein approximation (5.6), we obtain

Bq1,q2(p;x1,x2)p(x1,x2)=i=0n1j=0n2ai,j[Bq1,q2(x1ix2j;x1,x2)x1ix2j],
which implies, using (5.7),
Bq1,q2(p;x1,x2)p(x1,x2)=k=0q1l=0q2(i=0n1i=0n2ai,jδk,lq1,q2(i,j))bk,lq1,q2(x1,x2).(5.14)

Now, considering the form (5.3), we have

Bq1,q2(p;x1,x2)p(x1,x2)=k=0q1l=0q2(p(kq1,lq2)ck,lq1,q2)bk,lq1,q2(x1,x2).(5.15)

Equating the Bernstein coefficients of expressions (5.14) and (5.15), and using (5.9), we conclude that

p(kq1,lq2)=ck,lq1,q2+i=0n1j=0n2ai,jδk,lq1,q2(i,j)ck,lq1,q2+i=0n1j=0n2|ai,j|δk,lq1,q2(i,j)ck,lq1,q2+γ1(q11)q12+γ2(q21)q22,
from which follows the result.

From Theorems 5.2 and 5.3, it follows that c(q1,q2)λ as q1 and q2 and, therefore, Theorem 1.1 follows as a corollary.

6 CONCLUDING REMARKS

The representation of polynomials that are positive on the unit interval or any compact subset of n is an important subject with direct applications related to moment problems. See [5][5] J.B. Lasserre. “Moments, Positive Polynomials and their applications”, volume 1. Imperial College Press (2006). for more on this relation. The authors searched for the proof of Theorem 1.1 precisely to prove that the moment problem on the unit square has a solution—i.e. there is a finite representing measure for a sequence of moments—if and only if there is a positive linear functional for all polynomials that are nonnegative on the unit square. Not being aware of the work of Lasserre [5][5] J.B. Lasserre. “Moments, Positive Polynomials and their applications”, volume 1. Imperial College Press (2006)., where the result similar to the one we wanted to prove is demonstrated, we used the univariate results from Bernstein [1][1] S.N. Bernstein. Sur la représentation des polynômes positifs. Communications de la Société mathématique de Kharkow, 14(5) (1915), 227–228. and Hausdorff [4][4] F. Hausdorff. Summationsmethoden und momentfolgen. I. Mathematische Zeitschrift, 9(1-2) (1921), 74–109. as a stepping stone to build the proof for the unit square as described in Section 4.

Once our proof was concluded, we have found references [3][3] J. Garloff. Convergent bounds for the range of multivariate polynomials. In “International Symposium on Interval Mathematics”. Springer (1985), pp. 37–56. and [8][8] T. Rivlin. Bounds on a polynomial. Journal of Research of the National Bureau of Standards, 74 (1970), 47–57., which provided a demonstration for a similar result. Eventually we came across the book by Lasserre [5][5] J.B. Lasserre. “Moments, Positive Polynomials and their applications”, volume 1. Imperial College Press (2006)., where we found a theorem that is similar to Theorem 1.1, proved by Cassier [2][2] G. Cassier. Problème des moments sur un compact de rn et représentation de polynômes à plusieurs variables. Journal of Functional Analysis, 58 (1984), 256–266.. We briefly present such result, giving the formulation of [5][5] J.B. Lasserre. “Moments, Positive Polynomials and their applications”, volume 1. Imperial College Press (2006).. Let [𝐱]=[x1,,xn] be the ring of real multivariate polynomials and 𝕂 be a basic semi-algebraic set, subset of n

𝕂 := { 𝐱 n : g j ( 𝐱 ) 0 , j = 1 , , m } , (6.1)

where gj(𝐱)[𝐱], j=1,,m. Cassier [2][2] G. Cassier. Problème des moments sur un compact de rn et représentation de polynômes à plusieurs variables. Journal of Functional Analysis, 58 (1984), 256–266. has proven the following theorem.

Theorem 6.1. Let gj(𝐱)[𝐱] be affine for every j=1,,m and assume that 𝕂, as defined by (6.1), is compact with nonempty interior. If f[𝐱] is strictly positive on 𝕂 then

f = α m c α g 1 α 1 g m α m ,
for finitely many nonnegative scalars (cα).

If 𝐱=(x1,x2)2, g1(𝐱)=x1, g2(𝐱)=1x1, g3(𝐱)=x2 and g4(𝐱)=1x2, then 𝕂=[0,1]×[0,1]=I. When f is a positive polynomial on 𝕂 the theorem applies and there are nonnegative cα such that

f ( x 1 , x 2 ) = α 2 c α x 1 α 1 ( 1 x 1 ) α 2 x 2 α 3 ( 1 x 2 ) α 4 .

The main difference between the above Theorem and Theorem 1.1 is that the latter constructively derives the positive integers q1 and q2, the degrees of the Bernstein representation.

Both strategies developed in Sections 4 and 5 can be generalized to prove similar theorems for polynomials that are positive over arbitrary hypercubes.

REFERENCES

  • [1]
    S.N. Bernstein. Sur la représentation des polynômes positifs. Communications de la Société mathématique de Kharkow, 14(5) (1915), 227–228.
  • [2]
    G. Cassier. Problème des moments sur un compact de rn et représentation de polynômes à plusieurs variables. Journal of Functional Analysis, 58 (1984), 256–266.
  • [3]
    J. Garloff. Convergent bounds for the range of multivariate polynomials. In “International Symposium on Interval Mathematics”. Springer (1985), pp. 37–56.
  • [4]
    F. Hausdorff. Summationsmethoden und momentfolgen. I. Mathematische Zeitschrift, 9(1-2) (1921), 74–109.
  • [5]
    J.B. Lasserre. “Moments, Positive Polynomials and their applications”, volume 1. Imperial College Press (2006).
  • [6]
    G. Pólya. Über positive darstellung von polynomen. Vierteljschr. Naturforsch. Ges. Zürich, 73 (1928), 141–145.
  • [7]
    V. Powers & B. Reznick. Polynomials that are positive on an interval. Transactions of the American Mathematical Society, 352(10) (2000), 4677–4692.
  • [8]
    T. Rivlin. Bounds on a polynomial. Journal of Research of the National Bureau of Standards, 74 (1970), 47–57.

Publication Dates

  • Publication in this collection
    13 Dec 2019
  • Date of issue
    Sep-Dec 2019

History

  • Received
    18 June 2019
  • Accepted
    29 Aug 2019
Sociedade Brasileira de Matemática Aplicada e Computacional Rua Maestro João Seppe, nº. 900, 16º. andar - Sala 163 , 13561-120 São Carlos - SP, Tel. / Fax: (55 16) 3412-9752 - São Carlos - SP - Brazil
E-mail: sbmac@sbmac.org.br