Comput. Methods Funct. Theory https://doi.org/10.1007/s40315-018-0237-3

Complex Orthogonal Polynomials and Numerical Quadrature via Hyponormality David P. Kimsey1 · Mihai Putinar1,2

Received: 27 October 2016 / Revised: 29 August 2017 / Accepted: 26 December 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract By interpreting a truncated moment problem in two real variables in complex terms one exploits the hyponormality of the coordinate multiplier to obtain characterizations of: the moments of a positive measure, the associated complex orthogonal polynomials and the structure of the Hessenberg matrices. All reflecting the intricate structure of multivariate positive polynomials versus sums of squares or sums of hermitian squares of polynomials. The principal step in the proposed algorithm is stated in terms of a linear matrix inequality, making our approach prone for numerical applications. Keywords Truncated moment problem · Complex moment problem · Subnormal operator · Hyponormal operator · Quadrature formula · LMI · Multivariate Favard theorem Mathematics Subject Classification 44A60

Communicated by Laurent Baratchart.

B

David P. Kimsey [email protected] Mihai Putinar [email protected]; [email protected]

1

School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK

2

Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA 93106-3080, USA

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1 Introduction The foundations of an important area of constructive function theory lie at the interface between orthogonal polynomials, Gauss type quadratures, Jacobi matrices and the moment problem. While these objects and their relations are well understood on the line or the circle, when passing to two or more real variables inherent complications occur. We offer a novel approach to these algebraic aspects of constructive function theory in several variables by invoking a simple positivity criterion expressed solely in terms of extended moment data. Consider the following elementary questions: (A) Given complex polynomials p0 (z), p1 (z), . . . , pd (z) of exact degrees deg pk = k, 0 ≤ k ≤ d, when does there exist a positive measure μ on C, admitting all moments up to degree 2d, so that  p j (z) p  (z)dμ(z) = δ j , 0 ≤ j,  ≤ d? (B) Given a positive semidefinite (hermitian matrix) G = (s jk )dj,k=0 ∈ C(d+1)×(d+1) , when does there exist a positive measure μ on C such that  s jk =

z j z k dμ(z), 0 ≤ j, k ≤ d?

(C) Let μ be a positive measure on C admitting all moments and fix a positive integer d. Construct a positive measure with finite support ν, so that 

 f (z, z)dμ(z) =

f (z, z)dν(z),

for all polynomials f satisfying degz ( f ) ≤ d and degz ( f ) ≤ d. In spite of the well known classical solutions to the above questions in case supp μ ⊆ R or supp μ ⊆ T, a complete picture of the solutions in the case of two real variables is still far from being complete. Of course, the three questions are interlaced, as we will remind the reader a little later in this note. Problem (A) resonates with Favard’s Theorem on the line or on the circle. The heart of the matter is Problem (B). Given complex numbers (s jk )0≤ j,k≤d with s00 > 0 and s jk = sk j , the truncated moment problem on C entails determining whether or not there exists a positive Borel measure μ on C, with convergent moments, i.e.,  |z j z k |dμ(z) < ∞, 0 ≤ j, k < ∞, C



such that s jk =

123

C

z j z k dμ(z),

0 ≤ j, k ≤ d.

(1.1)

Complex Orthogonal Polynomials and Numerical Quadrature…

If μ satisfies (1.1), then μ is called a representing measure for (s jk )0≤ j,k≤d . Given a positive semidefinite matrix G = (s jk )dj,k=0 we offer a decision procedure for G to be the truncated moment matrix, in complex coordinates, of a positive measure in the complex plane. For a given matrix G denote by G jk the submatrix obtained from G by deleting the jth row and the kth column. Specifically, our main result states that, assuming det G = 0 and det G dd > 0, representation (1.1) holds true if and only if  00 d0  G G  0. (G) := G 0d G dd In addition, in this situation the measure μ is unique and has exactly d distinct atoms. The assumption G positive definite is also considered. In this case, a criterion for solubility is found in terms of the aforementioned result and can be communicated in terms of LMIs (linear matrix inequalities). The proof uses the hyponormality of a finite dimensional shift operator (which is equivalent to normality in this finite dimensional context). The hyponormality approach also admits a multivariate analogue, sketched in a separate section of this article. In sharp contrast to the one-dimensional case, the infinite moment matrix of a positive measure in R2 does not possess an effective characterization. Our algorithm necessarily involves matrix completions and equations with matrix unknowns, but these unknowns appear only linearly, in so called LMIs. In addition the sizes of the unknown entries in the completed matrices are simple functions of the rank of the original G. The truncated moment problem on C is usually formulated on total degree indexing sets of the form   ( j, k) ∈ N20 : 0 ≤ j + k ≤ d , where N20 denotes the set of all 2-tuples of non-negative integers. In the total degree setting, the flat extension approach of Curto and Fialkow originated in [3] and has been further investigated in a number of papers, e.g., [4–6]. In [11], a matrix analogue of the complex moment problem was considered for truncated sequences with a family of indexing sets that includes {( j, k):0 ≤ j, k ≤ d}\{(d, d)}. This approach was further developed in the scalar case in the paper [10]. For the relevance of the geometry of cut-off finite regions in the multivariate setting of the truncated moment problem and orthogonal polynomials see also the recent note [19]. It is well-known that the truncated moment problem on C is equivalent to the truncated moment problem on R2 in a natural way. There are some advantages to considering moment problems on C instead of R2 , see, e.g., [16] and [7,9]. Truncated moment problems have a number of applications, e.g., polynomial optimization, optimal control and mathematical finance (see, e.g., [12]) and optimal power flow (see, e.g., [9]). Let H be a Hilbert space with inner product ·, · and let B(H) denote the Banach space of bounded linear operators on H with the natural operator norm. An operator T ∈ B(H) is called normal and hyponormal if T ∗ T − T T ∗ = 0 and T ∗ T − T T ∗  0,

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i.e., (T ∗ T − T T ∗ )ξ, ξ  ≥ 0 for all ξ ∈ H. An operator S ∈ B(H) is called subnormal if there exists a Hilbert space K ⊇ H and a normal operator T ∈ B(K) such that T |H = S. It is readily checked that all subnormal operators are hyponormal. However, the converse does not hold. For more information on subnormal and hyponormal operators, see, e.g., [2,14,15]. It is worth noting that if H is finitely dimensional, then it follows from [2, Corr. 4.9] that T is normal ⇐⇒ T is hyponormal.

(1.2)

The equivalence in (1.2) seems not to have been used to address the truncated moment problem on C. We will use (1.2) to provide a solution criterion for the truncated moment problem on C and also a multivariate analogue on Cn . In the present note, we confine ourselves to the bare facts, leaving numerical experiments and ramifications of the main idea to be developed elsewhere.

2 Preliminaries Given complex numbers (s jk )0≤ j,k≤d with s00 > 0 and s jk = sk j , let ⎞ ⎛ s00 · · · sd0 ⎟ ⎜ G := ⎝ ... . . . ... ⎠ ∈ C(d+1)×(d+1) .

(2.1)

s0d · · · sdd

Lemma 2.1 If s = (s jk )0≤ j,k≤d has a representing measure μ and G is as in (2.1), then rank G ≤ card supp μ. (2.2) Proof If supp μ is infinite, then (2.2) obviously holds. If card supp μ is finite, say μ = nm=1 m δz m , where m > 0 and z 1 , . . . , z n are distinct complex numbers, then (2.2) follows from the factorisation G = V ∗ RV, where V ∈ Cn×(d+1) is the Vandermonde matrix whose entry in the row indexed by j m and column indexed by ( j, k) is given by z m z m k and R = diag(1 , . . . , n ).   We will also be in need of the following alternative formulation of a theorem of Bayer and Teichmann which is known as Tchakaloff’s theorem (since it generalises the main result of [18]). Theorem 2.2 If s = (s jk )0≤ j,k≤d has a representing measure μ, then there exists a representing measure ν with supp ν ⊆ supp μ and card supp ν ≤ (d + 1)2 .

(2.3)

Proof The proof can be carried out in much the same way as in [1] (see also [13]) by starting out with the complex vector

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Complex Orthogonal Polynomials and Numerical Quadrature…

v = col(s jk )0≤ j,k≤d . The bound given in (2.3) comes from the fact that v ∈ C(d+1) . 2

 

Lemma 2.3 Let A ∈ Cn×n be given. Suppose A  0. Then for any x ∈ Cn , there exists y ∈ C such that   A x  0 and det Aext = 0. Aext = (2.4) x∗ y Moreover, Aext  0 if and only if y > x ∗ A−1 x. Proof Since A is invertible, a Schur complement argument easily shows that Aext  0  if and only if y ≥ x ∗ A−1 x. Moreover, det Aext = 0 precisely when y = x ∗ A−1 x. 

3 Main Results Given (s jk )0≤ j,k≤d , let G be the matrix given by (2.1) and G jk denote the principal submatrix of G obtained by removing the jth row and kth column. We shall let C[z, z] and C[z] denote the ring of polynomials in z and z and z, respectively, with complex coefficients. We shall let Cd [z] and Cd [z, z] denote the set of polynomials in C[z] and C[z, z], respectively, with degree less than or equal to d. Theorem 3.1 Given complex numbers s = (s jk )0≤ j,k≤d , with d > 0, let G be as in (2.1) and suppose G  0, rank G = d and that the leading principal submatrix G dd is invertible. s has a representing measure μ if and only if  00 d0  G G  0. (3.1) (G) := G 0d G dd In this case, μ is unique and d-atomic. Remark 3.2 If the inequality (3.1) is in force, then (G) is hyponormal, or, equivalently, (G) is normal which forces rank (G) = d. Proof of Theorem 3.1 Suppose that μ is a representing measure for s. Since G is  := G dd is invertible, there exists a unique vector ξ = col(ξm )d−1 singular and G m=0 such that ⎛ ⎞ s0,d . ⎟  =⎜ Gξ ⎝ .. ⎠ . sd−1,d Thus, if P(z) = z d −

d−1

m=0 ξm z

m,

then

 C

|P(z)|2 dμ(z) = 0,

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which forces supp μ ⊆ {z ∈ C : P(z) = 0}.

(3.2)

Using Lemma 2.1, we actually have equality in (3.2). Thus,    L 2 (μ) := measurable f : | f (z)|2 dμ(z) < ∞ C

endowed with the natural inner product with respect to μ, has dimension d (and hence can be identified with Cd−1 [z]). Let T = Mz denote the operator of multiplication by z on L 2 (μ). It is readily checked that T is normal on L 2 (μ) and, hence, T is hyponormal. Thus,   ∗ T T T 0 on L 2 (μ) ⊕ L 2 (μ), T∗ I which is equivalent to (3.1).   0, there exists a set of linearly Conversely, suppose (3.1) is in force. Since G independent vectors w0 , . . . , wd−1 ∈ Cd such that    = w j , wk  d−1 , G j,k=0 where ·, · denotes the usual inner product in Cd (see, e.g., [8, Corollary 7.2.11]). Let T : Cd → Cd be given by T w j = w j+1 ,

j = 0, . . . , d − 2

and T wd−1 =

d−1 

sm,d wm .

m=0

Thus, T is cyclic. Moreover, as ⎛

0

⎜ ⎜ T = ⎜1 ⎝ 0

..

. 0 1

⎞ −s0,d ⎟ .. ⎟ . ⎟, ⎠ −sd−1,d

T is a companion matrix (see, e.g., [8, p. 146]) and hence the polynomial P(z) = m z d − d−1 m=0 sm,d z is the minimal polynomial for the cyclic operator T . Since G = (T j w0 , T k w0 )dj,k=0 , (3.1) is equivalent to the block matrix ⎛ ⎞ j+1 w , T k w )d−1 (T j+1 w0 , T k+1 w0 )d−1 0 0 j,k=0 j,k=0 (T ⎝ ⎠  0. d−1 d−1 j k+1 j k (T w0 , T w0 ) j,k=0 (T w0 , T w0 ) j,k=0

123

(3.3)

Complex Orthogonal Polynomials and Numerical Quadrature…

Since T j+1 w0 , T k+1 w0  = T ∗ T w j , wk , T j+1 w0 , T k w0  = T w j , wk  and T j w0 , T k w0  = w j , wk , (3.3) is equivalent to     ∗ ∗ T ∗ T T ξ ξ η ≥ 0, T∗ I η

ξ, η ∈ Cd .

Thus,   ∗ T T T   0, on Cd ⊕ Cd . T = T∗ I   0 if and only if It follows from a Schur complement argument that T T ∗ T − T T ∗  0, i.e., T is hyponormal. Since T : Cd → Cd , hyponormality is equivalent to normality (see, e.g., [2, Corollary 4.9]). Thus, the spectral theorem (see, e.g., [8, Theorem 2.5.4]) asserts the existence of points λ1 , . . . , λd ∈ C and mutually orthogonal projections Pλ1 , . . . , Pλd such that T =

d 

λm Pλm .

m=1

Therefore, s jk = T j (T ∗ )k w0 , w0  =

d 

j

k

λm λm Pλm w0 , w0 , 0 ≤ j, k ≤ d.

m=1

If we let m = Pλm w0 , w0 , then μ = dm=1 m δλm is a representing measure for s with at most d-atoms. Inview of Lemma 2.1, μ must have exactly d atoms. Finally, let Q(z) = dm=1 (z − λm )m be the unique monic polynomial in Cd [z] with the prescribed zeroes {λ1 , . . . , λd }. Suppose ν is another representing measure for (s jk )0≤ j,k≤d . Then 

 z z dμ(z) = j k

C

C

z j z k dν(z), 0 ≤ j, k ≤ d,

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forces 

 C

Since

 C

Q(z)dμ(z) =

Q(z)dμ(z) = 0, we must have

 C

C

Q(z)dν(z).

Q(z)dν(z) = 0 as well, whence

supp ν ⊆ {λ1 , . . . , λd }.

(3.4)

Lemma 2.1 forces equality in (3.4). Thus, μ is the unique representing measure for (s jk )0≤ j,k≤d .   In the following result, we will make use of the notation  

(m) := ( j, k) ∈ N20 : 0 ≤ j, k ≤ m . Corollary 3.3 Given complex numbers s = (s jk )0≤ j,k≤d , let G be as in (2.1) and suppose G  0. s has a representing measure μ if and only if there exist (s jk )( j,k)∈ (κ) \ (d) such that, for some κ ∈ {d + 1, . . . , d(d + 1) + 1},  C=

G X X Y∗



= (s jk )κj,k=0  0

and has the following properties: (i) det C = 0 and C κ,κ  0. (ii) (C)  0 [see (3.1)]. Proof In view of Theorem 2.2, s has a representing measure if and only if s has a representing measure ν with card supp ν ≤ (d + 1)2 . The equivalence of s having a representing measure and properties (i) and (ii) follow from repeated applications of Lemma 2.3 and Theorem 3.1.   Remark 3.4 If κ = d + 1, then conditions (i) and (ii) in Corollary 3.3 are equivalent to solving the following LMI: ξ ∗ G 00 ξ + η∗ Gη + |ξd |2 x ∗ G −1 x + 2Re(η∗ G d0 ξ + xd ηd ξd ) ⎛ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞T ⎞ x1 x0 sd1 ⎜ ∗⎜ . ⎟ ⎜ ⎟ ⎜ . . ⎟ ⎟ ∗ ⎟ + 2Re ⎜ ⎝ξd ξ ⎝ .. ⎠ + ξd η ⎝ .. ⎠ + ηd ⎝ .. ⎠ ξ ⎠ ≥ 0, xd

xd−1

sdd

If d = 1 and x0 := s02 , x1 := s12 and y := s22 , then (3.5) can be written as η∗ Gη + |ξ0 |2 s11 + |ξ1 |2 y + 2Re(x1 ξ1 ξ0 ) + 2Re(x0 ξ0 ξ3 ) + 2Re(x0 ξ1 ξ2 ) + 2Re(x1 ξ1 ξ3 ) ≥ 0,

123

(3.5)

Complex Orthogonal Polynomials and Numerical Quadrature…

where  y=

1 s00 s11 − |s01 |2

  |x0 |2 s11 − 2Re(s01 x1 + |x1 |2 s00 .

Example 3.5 In the following example we will use Corollary 3.3 to compute an infinite family of d + 2-atomic representing measures for the following truncated sequence. Let μ denote the planar measure on D and μ0 = π1 μ. Put  s jk =

 =

z j z k dμ0 (z)

C

1 j+k

0

if

j =k

if

j = k, 0 ≤ j, k ≤ d.

Then, ⎛

s00 · · · ⎜ .. . . G=⎝ . . s0d · · ·

⎞ sd0 .. ⎟ = diag(1, 1/2, . . . , 1/(d + 1))  0. . ⎠ sdd

Moreover, ⎛ ⎛ ⎜ C =⎝

s00 · · · .. . . . . s0,d+1 · · ·

1

0

x0

⎞

⎜ ⎟ ⎜0 1 0 x1 ⎟ 2 ⎜ ⎟ sd+1,0 ⎜ ⎟ ⎟ ⎜ . .. .. .. ⎟ ⎟ ⎠=⎜ . . ⎜ ⎟ ⎜ ⎟ sd+1,d+1 1 ⎜0 ⎟ ⎝ d+1 x d ⎠ x0 · · · xd xd+1 ⎞

is positive semidefinite and singular if and only if   ∗  xd+1 = x0 · · · xd G −1 x0 · · · xd . Thus, if x1 = · · · = xd = 0, then xd+1 = |x0 |2 . In this case, using nested determinants, it is easy to see that  (C) =

 A B , B∗ D

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where A = diag(1/2, . . . , 1/(d + 1), |x0 |2 ) ⎛ ⎞ 0 x0 ⎜1 .. ⎟ ⎜ . 0⎟ 2 ⎜ ⎟ B= ⎜ . .. ⎟ ⎝ .. .⎠ 1 0 d+1 0 and D = diag(1, 1/2, . . . , 1/(d + 1)). Since C  0, it follows from a Schur complement argument that (C)  0 if and only if A − B D −1 B ∗  0. However, it is readily checked that A − B D −1 B ∗ = diag



 d −1 1 1 d 2 − (d + 1)|x0 |2 , . . . , − , , |x | − 0 2 d +1 d2 d +1

which is obviously positive semidefinite if and only if √ d |x0 | ≥ . d +1 Thus, a (d + 1)-atomic representing measure for (s jk )0≤ j,k≤d is given by

ν=

d+1 

q δzq ,

q=1

where z q are the distinct zeros of the polynomial z d+1 − x0 and the weights are given by ⎞−1 ⎛ ⎞ ⎛ ⎞ 1 ··· 1 1 1 ⎜ z 1 · · · z d+1 ⎟ ⎜0⎟ ⎜ .. ⎟ ⎜ ⎟ ⎜ ⎟ .. ⎟ ⎜ .. ⎟ . ⎝ . ⎠ = ⎜ .. ⎝ . . ⎠ ⎝.⎠ d+1 0 z 1 d · · · z d+1 d ⎛

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Complex Orthogonal Polynomials and Numerical Quadrature…

4 Generalization to Several Variables Let Nn0 denote the set of all n-tuples of non-negative integers and  Idn

= (γ1 , . . . , γn ) ∈

Nd0

:0≤

n 

 γm ≤ d .

m=1

Given s = (sγ ,η )γ ,η∈Idn , the truncated moment problem on Cn entails determing whether or not there exists a positive Borel measure μ on Cn , with convergent moments, i.e.,  Cn

|z γ z η |dμ(z) < ∞, γ , η ∈ dn ,

where 

γ η

Cn



z z dμ(z) :=

 ···

n  Cn m=1

γ

z mm z m ηm dμ(z 1 , . . . , z n )

and γ = (γ1 , . . . , γn ) and η = (η1 , . . . , ηn ). Let G denote the complex matrix with columns indexed by the monomials {Z γ :Idn } and rows indexed by the monomials γ {Z : Idn } (ordered in a graded lexicographical fashion) with the entry in the row indexed γ γ γ by Z := Z 11 · · · Z nn and the column indexed by Z η := Z η1 · · · Z ηn be given by sγ ,η ,

γ , η ∈ Idn .

γ

Given non-empty subsets E ⊆ {Z :Idn } and F ⊆ {Z γ :Idn }, we will let G(E, F) be the submatrix of G with rows indexed by E and columns indexed by F. For example, if d = n = 2, then ⎛

s0000 ⎜s0010 ⎜ ⎜s0001 G=⎜ ⎜s0020 ⎜ ⎝s0011 s0002

s1000 s1010 s1001 s1020 s1011 s1002

s0100 s0110 s0101 s0120 s0111 s0102

s2000 s2010 s2001 s2020 s2011 s2002

s1100 s1120 s1101 s1120 s1111 s1102

⎞ s0200 s0210 ⎟ ⎟ s0201 ⎟ ⎟ s0220 ⎟ ⎟ s0211 ⎠ s0202

and ⎛ s0000 G({1, Z 1 , Z 2 }, {1, Z 1 , Z 2 }) = ⎝s0010 s0001

s1000 s1010 s1001

⎞ s0100 s0110 ⎠ . s0101

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Theorem 4.1 Given complex numbers s = (sγ ,η )γ ,η∈Idn , with d > 0, let G be as above and suppose G  0,    = G {Z γ }γ ∈I n , {Z γ }γ ∈I n 0 G d−1 d−1  Put and rank G = rank G. −1  γ   γ n n . Tz m = G {Z γ }γ ∈Ind−1 , {Z }γ ∈Id−1 G {Z }γ ∈Ind−1 , {Z m Z γ }γ ∈Id−1

(4.1)

s has a representing measure μ if and only if Tz m Tza = Tza Tz m ,

1 ≤ a, m ≤ n

(4.2)

and ⎛    γ ⎞ γ n , {Z m Z γ }γ ∈I n n , {Z m Z γ }γ ∈I n G {Z m Z }γ ∈Id−1 G {Z } γ ∈ I d−1 d−1    γ d−1  ⎠ m (G) := ⎝ γ n , {Z γ }γ ∈I n n , {Z γ }γ ∈I n G {Z m Z }γ ∈Id−1 G {Z } γ ∈ I d−1 d−1 d−1  is positive semidefinite for m = 1, . . . , n. In this case, μ is unique and (rank G)atomic. Proof The positivity of m (G) is equivalent to the hyponormality of Tz m which in turn is equivalent to the normality of Tz m . In view of (4.1), Tz 1 , . . . , Tz m is a family of commuting normal operators. The rest of the proof follows in much the same way as the proof of Theorem 3.1.  

5 Final Remarks Having completed the proofs of our decision procedure, it is time to look back from this perspective at the problems stated in the introduction. We confine us to discuss only the 2D case, that is one complex variable. Let p0 (z), p1 (z), . . . , pd (z) of exact degrees deg pk = k, 0 ≤ k ≤ d, be complex polynomials as in Problem A. Due to the independence of this family, one can write zk =

k 

L k j p j (z),

j=0

with some complex coefficients L k j . Note that the matrix L = (L k j )dk, j=0 is lower triangular and invertible.

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Complex Orthogonal Polynomials and Numerical Quadrature…

Assume that a positive measure μ exists which turns p0 (z), p1 (z), . . . , pd (z) into an orthonormal family. Then the moment matrix G appearing in Problem B satisfies z j , z k μ =

j  k 

L k j L nk δmn ,

m=0 n=0

that is G = L L ∗, which is nothing but the Cholesky’s LU decomposition of the invertible matrix G. Thus, Problem A is contained rather trivially in Problem B. However, this does not illuminate why not all sequences of polynomials can be orthonormalized by a positive measure. Or equivalently, why not all positive matrices are moment matrices (as G above) of a positive measure. The departure from the one variable case and the explanation lies in the fact that not all positive polynomials are sums of squares. A fact already known to Hilbert. Lets look closer to this obstruction. Let G = (s j k) be a positive definite (d + 1) × (d + 1) complex matrix, indexed as above from 0 to d. This matrix defines a non-degenerate inner product on the vector space Cd [z] of polynomials of degree less than or equal to d: z j , z  G = s jk , 0 ≤ j, k ≤ d. Or equivalently, a linear functional λG :Cd,d [z, z] −→ C is given so that λG (| p(z)|2 ) ≥ 0, for every p ∈ Cd [z]. That is λG is real, i.e. commutes with the conjugation, and is non-negative on sums of hermitian squares of polynomials, denoted henceforth h2 . If G were the moment matrix of a positive measure, then we would have λG ( f (z, z)) ≥ 0 for all non-negative polynomials f . And this is not always the case. Indeed, 4(x − y)2 + 1 = [(1 + i)z + (1 − i)z]2 + 1 is a positive polynomial which cannot be written as a sum of squares of real polynomials. Just monitor the highest degree terms! However, 4(x − y)2 + 1 ∈  2 , that is this polynomial is a sums of squares of real polynomials. It is not hard to see that the convex cones h2 ∩ Cd,d [z, z] ⊂  2 ∩ Cd,d [z, z] are closed and distinct (for d ≥ 2) in the topology of the vector space of polynomials. By the Minkowski separation theorem, there exists a separating functional λG which

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is positive on the smaller cone but not on the larger one. That is there exists a positive matrix G which is not the moment matrix of a positive measure on C. And worse than that, there are positive polynomials which are not sums of squares, see, for instance [13]. The subtle difference between sums of hermitian squares versus sums of squares, along real algebraic varieties, is treated in the recent article [17]. Returning to Problem B, our dichotomy gives the following. Assume G is positive semidefinite of type (d + 1) × (d + 1). Let [G]k denote the principal minor of order k of G, that is, [G]k = (si j )0≤i, j 0 and (C) ≥ 0. Obviously this also solves Problem C. The alert reader may ask what is the analogue of the Jacobi matrix in this story. It is a Hessenberg matrix representing the multiplication by the complex variable on the space Cd [z], and our main result is nothing else than a normality criterion for this matrix. The computational complexity of this normality criterion is however much higher than the positivity of our matrix . To be very precise, start with the positive semidefinite matrix G = (s jk ) of type (d + 1) × (d + 1) satisfying det G = 0 and det G dd > 0. We define an inner product on the space of polynomials Cd [z] by z j , z k  = s jk , 0 ≤ j, k ≤ d. This hermitian form is non-degenerate on Cd−1 [z] but there exists a vector pd ∈ Cd [z]\Cd−1 [z] which has length zero:  pd , pd  = 0.

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Complex Orthogonal Polynomials and Numerical Quadrature…

Let p0 , p1 , . . . , pd−1 be an orthonormal base of Cd−1 [z] consisting of polynomials of exact degree deg pk = k, 0 ≤ k < d. To obtain this numerically we split G dd = L L ∗ into LU form and invert the lower triangular matrix L. Specifically: ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

1 z z2 .. .

z d−1

⎞

⎛

⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎠ ⎝

L 00 L 10 L 20 .. .

0 L 11 L 21

0 ... 0 ... L 22 0 .. .

L d−1,0 L d−1,1 . . .

⎞⎛

0 0 ...

⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎠⎝

L d−1,d−1

p0 (z) p1 (z) p2 (z) .. .

⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎠

pd−1 (z)

Once the basis p j (z) is found one can represent the multiplication by the variable as zpk (z) =

k+1 

h jk p j (z), 0 ≤ j < d − 1.

j=0

For the highest degree polynomial we define the coefficients h j,d−1 by the property zpd−1 −

d−1 

h j,d−1 p j (z) = cpd (z),

j=0

where c = 0. The resulting Hessenberg matrix ⎛

h 00 h 01 ⎜ h 10 h 11 ⎜ ⎜ H = ⎜ 0 h 21 ⎜ .. ⎝ . 0

...

h 02 h 12 h 22 .. .

... ... ...

h 0,d−1 h 1,d−1 h 2,d−1 .. .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

h d−2,d−1 h d−1,d−1

represents in the basis ( p0 , p1 , . . . , pd−1 ) the multiplication by the variable z. By repeating the proof of our main result we deduce that G is the moment matrix of a positive measure if and only if the matrix H is normal, that is H H ∗ = H ∗ H. This observation gives an alternative route to check the principal step of our dichotomy. We mention in the spirit of the present note that hyponormal operators have successfully been exploited during several decades in treating a Markov type moment problem in two dimensions. Namely the moments of a “shade function” of compact support

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g:C −→ [0, 1] can effectively be stored and processed by an irreducible hyponormal operator with rank-one self-commutator. See [14, Section XII.3] for an early account of the theory. Within this framework the class of planar domains carrying a finite point cubature formula for all complex analytic functions, the so called “quadrature domains”, turned out to coincide with the extremal solutions of the Markov moment problem. In addition, the study of the asymptotics of the associated orthogonal polynomials, arising as characteristic polynomials of the finite central truncations of the respective hyponormal operator, opened a whole new chapter of limit behaviors of potential theory flavor. See [7] for the latter and a recent survey of the theory.

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