Positive Polynomials on closed boxes

We present two different proofs that positive polynomials on closed boxes of $\mathbb{R}^2$ 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 $\mathbb{R}^n$, $n>2$.


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: and, for every (x 1 , x 2 ) ∈ I, p(x 1 , x 2 ) > 0, then there exist q 1 ≥ n 1 , q 2 ≥ n 2 and C i, j > 0, (i, j) ∈ Q 1 × Q 2 , such that where Q 1 = {0, 1, . . ., q 1 } and Q 2 = {0, 1, . . ., q 2 }.
Furthermore, we constructively derive the values of q 1 and q 2 . Theorem 1 is an extension of similar results obtained for positive polynomials on compact intervals and multidimensional simplexes by, respectively, Bernstein [1], Hausdorff [5] and Pólya [7]. We are aware that, using a different proof strategy, Cassier [2] has proven a general result from which a similar version of Theorem 1 follows. We discuss this more extensively at the final section.
We provide two proofs of Theorem 1. The first one is supported by results for the univariate version of Theorem 1, proved by Powers and Reznick [8]. The second proof extends the approach in Garloff [3] and Rivlin [9].
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, given in Section 4. Section 5 brings an alternative proof, based on [3] and [9].
2 Definitions and notation Definition 1. Let P n be the linear space of polynomials of degree n, i.e.
Definition 2. For any p ∈ P n we define its Goursat transformp bỹ Definition 3. Let B + n be the set of polynomials of degree n that can be written with non-negative coordinates in the Bernstein basis, Similarly, let B +, * n be the set of polynomials of degree n that can be written with positive coordinates in the Bernstein basis, Definition 4. For every a = (a 1 , . . ., a n ) ∈ R n , m ≥ n and 0 ≤ i ≤ m, let Definition 5. For every a = (a 1 , . . ., a n ) ∈ R n , let Notice that B k (a) is a linear combination of a.
Proof. Applying the Binomial theorem to the identity From this expression, we obtain that The proof that the A i 's are unique follows from observing that, The following theorem is a consequence of Theorem 6 in [8]. 1] p(x) and e j be such thatp( Proof. It follows from Theorem 2 that there exist A i ≥ 0 such that Observe that, for every k, ∑ Lemma 3. If p(x) = ∑ n i=0 a i x i and a = (a 1 , . . ., a n ), theñ The main idea behind this proof is to use twice the positive representation result for univariate polynomials (lemma 2). 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 x 2 ∈ [0, 1], obtain from definition 7 that 1]. From this observation, one can obtain two facts. First, since I is compact, Second, it follows from Lemma 3 that Since each B i is a linear combination of the elements of a(x 2 ) and each element of a(x 2 ) is a polynomial on Therefore, it follows from Lemma 2 and Equations (3) and (4) that, taking It follows from Lemma 2 that, taking q 2 = 2 3n 2 + max i one obtains that where C i, j > 0. By applying Equation (6) to Equation (5), one obtains

Alternative proof
We consider, as before, the bivariate polynomial p given in (1) and λ = inf (x 1 ,x 2 )∈I p(x 1 , x 2 ). For q 1 , q 2 ≥ 1, let us define the bivariate polynomial where k ∈ Q 1 and l ∈ Q 2 . The set of polynomials {b are the Bernstein polynomials of degree q 1 and q 2 and form a basis for the linear space of all bivariate polynomials of the form (1) with n 1 = q 1 and n 2 = q 2 .
Lemma 4. If i ∈ Q 1 and j ∈ Q 2 , then where it is assumed that m v = 0 for integers m and v such that m < v. Proof. The result follows by applying the Binomial theorem to the identity Henceforth, we shall consider q 1 ≥ n 1 , q 2 ≥ n 2 . Then, it follows from Lemma 4 that p(x 1 , x 2 ) given in (1) can be rewritten as where c q 1 ,q 2 k,l The c (q 1 ,q 2 ) k,l are the Bernstein coefficients and (9) is the Bernstein form of p(x 1 , x 2 ). In the sequel, we denote by Theorem 3. If p is given by (1), then for all (x 1 , x 2 ) ∈ I, which implies the assertion.
Theorem 4. If p is given by (1), q 1 ≥ n 1 and q 2 ≥ n 2 , then Proof. For any real function f (x 1 , x 2 ), define its Bernstein approximation on I by For 0 ≤ i ≤ n 1 and 0 ≤ j ≤ n 2 , let δ q 1 ,q 2 k,l (i, j), (k, l) ∈ Q 1 × Q 2 , be the Bernstein coefficients of the polynomial B q 1 , Then, from Lemma 4 and (12) , it follows that k ∈ Q 1 , l ∈ Q 2 . For any fixed 0 ≤ i ≤ n 1 and 0 ≤ j ≤ n 2 , we can prove that for all k ∈ Q 1 and l ∈ Q 2 . In order to prove (15), it suffices to show that Since (17) is essentially the same as (16), we only present the proof of (16). Notice that (16) clearly holds for i = 0, i = 1, k = 0 and k = q 1 . Thus, let us consider 1 ≤ k ≤ q 1 − 1 and i ≥ 2.

Concluding remarks
The representation of polynomials that are positive on the unit interval or any compact subset of R n is an important subject with direct applications related to moment problems. See [6] for more on this relation. The authors searched for the proof of theorem 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 [6], where the result similar to the one we wanted to prove is demonstrated, we used the univariate results from Bernstein [1] and Hausdorff [5] 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] and [9], which provided a demonstration for a similar result. Eventually we came across the book by Lasserre [6], where we found a theorem that is similar to Theorem 1, proved by Cassier [2]. We briefly present such result, giving the formulation of [6]. Let R[x] = R[x 1 , . . . , x n ] be the ring of real multivariate polynomials and K be a basic semi-algebraic set, subset of R n K := {x ∈ R n : g j (x) ≥ 0, j = 1, . . . , m} where g j (x) ∈ R[x], j = 1, . . . , m. Cassier [2] has proven the following theorem.
Theorem 5. Let g j (x) ∈ R[x] be affine for every j = 1, . . ., m and assume that K, as defined by (22), is compact with nonempty interior. If f ∈ R[x] is strictly positive on K then for finitely many nonnegative scalars (c α ).
If x = (x 1 , x 2 ) ∈ R 2 , g 1 (x) = x 1 , g 2 (x) = 1−x 1 , g 3 (x) = x 2 and g 4 (x) = 1−x 2 , then K = [0, 1] × [0, 1] = I. When f is a positive polynomial on K the theorem applies and there are nonnegative c α such that The main difference between the above Theorem and Theorem 1 is that the latter constructively derives the positive integers q 1 and q 2 , 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.