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Truncated Moment Problems:
The Extremal Case(joint work with L. Fialkow and M. Moller)
Raul E. Curto, University of Iowa
IWOTA 2006
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 1 / 58
Abstract
For a degree 2n real d-dimensional multisequence
β ≡ β(2n) = {βi}i∈Zd+,|i |≤2n to have a representing measure µ, it is
necessary for the associated moment matrix M(n)(β) to be positive
semidefinite, and for the algebraic variety associated to β, V ≡ Vβ, to
satisfy rankM(n) ≤ cardV as well as the following consistency
condition: if a polynomial p(x) ≡∑
|i |≤2n aixi vanishes on V, then
p(β) :=∑
|i |≤2n aiβi = 0.
In joint work with Lawrence Fialkow and Michael Moller, we prove
that for the extremal case (rankM(n) = cardV), positivity of M(n)
and consistency are sufficient for the existence of a (unique,
rankM(n)-atomic) representing measure.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 2 / 58
Abstract
The new results build on our operator-theoretic approach to truncated
moment problems, based on matrix positivity and extension, which,
via a “functional calculus” for the columns of the associated moment
matrix, allows us to obtain existence theorems in case the columns
satisfy one of several natural constraints.
[email protected]; http://www.math.uiowa.edu/˜rcurto
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 3 / 58
TCMP
Given γ : γ00, γ01, γ10, . . . , γ0,2n, . . . , γ2n,0, with γ00 > 0 and γji = γij ,
the TCMP entails finding a positive Borel measure µ supported in
the complex plane C such that
γij =
∫z iz jdµ (0 ≤ i + j ≤ 2n);
µ is called a rep. meas. for γ.
In earlier joint work with L. Fialkow,
We have introduced an approach based on matrix positivity and
extension, combined with a new “functional calculus” for the columns
of the associated moment matrix.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 4 / 58
We have shown that when the TCMP is of flat data type, a solution
always exists; this is compatible with our previous results for
supp µ ⊆ R (Hamburger TMP)
supp µ ⊆ [0,∞) (Stieltjes TMP)
supp µ ⊆ [a, b] (Hausdorff TMP)
supp µ ⊆ T (Toeplitz TMP)
Along the way we have developed new machinery for analyzing
TMP’s in one or several real or complex variables. For simplicity,
in this talk we focus on one complex variable or two real variables,
although several results have multivariable versions.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 5 / 58
Our techniques also give concrete algorithms to provide finitely-atomic
rep. meas. whose atoms and densities can be explicitly computed.
We have fully resolved, among others, the cases
Z = α1 + βZ
and
Z k = pk−1(Z , Z ) (1 ≤ k ≤ [n
2] + 1; deg pk−1 ≤ k − 1).
We obtain applications to quadrature problems in numerical analysis.
We have obtained a duality proof of a generalized form of the
Tchakaloff-Putinar Theorem on the existence of quadrature rules for
positive Borel measures on Rd .
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 6 / 58
Very recently, we have begun to use our methods to solve FULL
moment problems, by first solving truncated MP’s, and then applying
J. Stochel’s limiting argument.
Our matrix extension approach works equally well to localize the
support of a rep. meas.
In the specific case of K := supp µ, a semi-algebraic set determined
by a finite collection of complex polynomials P = {pi (z , z)}mi=1, i.e.,
K = KP := {z ∈ C : pi (z , z) ≥ 0, 1 ≤ i ≤ m},
we obtain an existence criterion expressed in terms of positivity and
extension properties of the moment matrix M (n) (γ) associated to γ
and of the localizing matrix Mpi corresponding to each pi .
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 7 / 58
Positivity of Block Matrices
Theorem
(Smul’jan, 1959)
(A B
B∗ C
)≥ 0 ⇔
A ≥ 0
B = AW
C ≥ W ∗AW
.
Moreover, rank
(A B
B∗ C
)=rank A ⇔ C = W ∗AW .
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 8 / 58
Corollary
Let A ≥ 0 and assume rank
(A B
B∗ C
)=rank A. Then(
A B
B∗ C
)≥ 0.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 9 / 58
Basic Positivity Condition
Pn : polynomials p in z and z , deg p ≤ n
Given p ∈ Pn, p(z , z) ≡∑
0≤i+j≤n aij ziz j ,
0 ≤∫| p(z , z) |2 dµ(z , z)
=∑ijk`
aij ak`
∫z i+`z j+kdµ(z , z)
=∑ijk`
aij ak`γi+`,j+k .
To understand this “matricial” positivity, we introduce the following
lexicographic order on the rows and columns of M(n):
1,Z , Z ,Z 2, ZZ , Z 2, . . .
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 10 / 58
Define M[i , j ] as in
M[3, 2] :=
γ32 γ41 γ50
γ23 γ32 γ41
γ14 γ23 γ32
γ05 γ14 γ23
Then
(“matricial” positivity)∑ijk`
aij ak`γi+`,j+k ≥ 0
⇔ M(n) ≡ M(n)(γ) :=
M[0, 0] M[0, 1] ... M[0, n]
M[1, 0] M[1, 1] ... M[1, n]
... ... ... . . .
M[n, 0] M[n, 1] . . . M[n, n]
≥ 0.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 11 / 58
For example,
M(1) =
γ00 γ01 γ10
γ10 γ11 γ20
γ01 γ02 γ11
,
M(2)=
γ00 γ01 γ10 γ02 γ11 γ20
γ10 γ11 γ20 γ12 γ21 γ30
γ01 γ02 γ11 γ03 γ12 γ21
γ20 γ21 γ12 γ22 γ31 γ40
γ11 γ12 γ21 γ13 γ22 γ31
γ02 γ03 γ12 γ04 γ13 γ22
.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 12 / 58
In general,
M(n + 1) =
(M(n) B
B∗ C
)Positivity Condition is not sufficient:
By modifying an example of K. Schmudgen, we have built a family
γ00, γ01,γ10, ..., γ06, ..., γ60 with positive invertible moment matrix M(3)
but no rep. meas. But this can also be done for n = 2.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 13 / 58
Moment Problems and Nonnegative
Polynomials (FULL MP Case)
M := {γ ≡ γ(∞) : γ admits a rep. meas. µ}
B+ := {γ ≡ γ(∞) : M(∞)(γ) ≥ 0}Clearly, M⊆ B+
(Berg, Christensen and Ressel) γ ∈ B+, γ bounded ⇒ γ ∈M
(Berg and Maserick) γ ∈ B+, γ exponentially bounded ⇒ γ ∈M
(RC and L. Fialkow) γ ∈ B+, M(γ) flat ⇒ γ ∈M
P+ : nonnegative poly’s
Σ2 : sums of squares of poly’s
Clearly, Σ2 ⊆ P+
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 14 / 58
Duality
For C a cone in RZn+ , we let
C ∗ := {x ∈ RZn+ : supp(x) is finite and 〈p, x〉 ≥ 0 for all p ∈ C}.
(Haviland) P∗+ = MFor, the Riesz functional Λγ(p) := p(γ) maps M→ P∗+ (γ 7→ Λγ),
and Haviland’s Theorem says that this maps is onto, that is, there
exists µ r.m. for γ if and only if Λγ ≥ 0 on P+.
At present, we are attempting to formulate and prove a version of this
result for TMP.
P+ = M∗ (straightforward once we have a r.m.)
B+ = (Σ2)∗ (straightforward)
(Berg, Christensen and Jensen) (B+)∗ = Σ2
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 15 / 58
(n = 1) P+ = Σ2 ⇒ P∗+ = (Σ2)∗ ⇒M = B+ (Hamburger)
(Hilbert) Description of pairs (n, d) for which every poly of degree d
in n indeterminates which is nonnegative on Rn is a sum of squares
(cf. Reznick 2000)
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 16 / 58
Functional Calculus
For p ∈ Pn, p(z , z) ≡∑
0≤i+j≤n aij ziz j define
p(Z , Z ) :=∑
aij ZiZ j .
If there exists a rep. meas. µ, then
p(Z , Z ) = 0 ⇔ supp µ ⊆ Z(p).
The following is our analogue of recursiveness for the TCMP
(RG) If p, q, pq ∈ Pn, and p(Z , Z ) = 0,
then (pq)(Z , Z ) = 0.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 17 / 58
First Existence Criterion
Theorem
(RC-LF, 1998) Let γ be a truncated moment sequence. TFAE:
(i) γ has a rep. meas.;
(ii) γ has a rep. meas. with moments of all orders;
(iii) γ has a compactly supported rep. meas.;
(iv) γ has a finitely atomic rep. meas. (with at most (n + 2)(2n + 3)
atoms);
(v) M(n) ≥ 0 and for some k ≥ 0 M(n) admits a positive extension
M(n + k), which in turn admits a flat (i.e., rank-preserving) extension
M(n + k + 1) (here k ≤ 2n2 + 6n + 6).
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 18 / 58
Case of Flat Data
Recall: If µ is a rep. meas. for M(n), then rank M(n) ≤ card supp µ.
γ is flat if M(n) =
(M(n − 1) M(n − 1)W
W ∗M(n − 1) W ∗M(n − 1)W
).
Theorem
(RC-LF, 1996) If γ is flat and M(n) ≥ 0, then M(n) admits a unique flat
extension of the form M(n + 1).
Theorem
(RC-LF, 1996) The truncated moment sequence γ has a
rank M(n)-atomic rep. meas. if and only if M(n) ≥ 0 and M(n) admits a
flat extension M(n + 1).
To find µ concretely, let r :=rank M(n) and look for the relation
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 19 / 58
Z r = c01 + c1Z + ... + cr−1Zr−1.
We then define
p(z) := z r − (c0 + ... + cr−1zr−1)
and solve the Vandermonde equation1 · · · 1
z0 · · · zr−1
· · · · · · · · ·z r−10 · · · z r−1
r−1
ρ0
ρ1
· · ·ρr−1
=
γ00
γ01
· · ·γ0r−1
.
Then
µ =r−1∑j=0
ρjδzj .
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 20 / 58
Localizing Matrices
Consider the full MP ∫z iz j dµ = γij (i , j ≥ 0),
where supp µ⊆ K , for K a closed subset of C.
The Riesz functional is given by
Λγ(z iz j) := γij (i , j ≥ 0).
Riesz-Haviland: There exists µ with supp µ ⊆ K ⇔ Λγ(p) ≥ 0 for all
p such that p|K ≥ 0.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 21 / 58
If q is a polynomial in z and z , and
K ≡ Kq := {z ∈ C : q(z , z) ≥ 0},
then Lq(p) := L(qp) must satisfy Lq(pp) ≥ 0 for µ to exist. For,
Lq(pp) =
∫Kq
qpp dµ ≥ 0 (all p).
K. Schmudgen (1991): If Kq is compact, Λγ(pp) ≥ 0 and
Lq(pp) ≥ 0 for all p, then there exists µ with supp µ ⊆ Kq.
We shall establish a version of this result for truncated MP’s.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 22 / 58
First, recall that p(Z , Z ) = 0 implies supp µ ⊆ Z(p). We define the
algebraic variety of γ as
V(γ) :=⋂
p∈Pn
p(Z ,Z)=0
Z(p),
and observe that rank M(n) ≤ card supp µ ≤ card V (γ), from which it
follows that
card V(γ) < rank M(n) ⇒ there is no rep. meas. µ.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 23 / 58
Localization of Support: Main Theorem
Theorem
(RC-LF, 2000) Let M(n) ≥ 0 and suppose deg(q) = 2k or 2k − 1 for
some k ≤ n. Then ∃ µ with rank M(n) atoms and supp µ ⊆ Kq if and
only if ∃ a flat extension M(n + 1) for which Mq(n + k) ≥ 0. In this case,
∃ µ with exactly rank M(n)− rank Mq(n + k) atoms in Z(q).
Remark
M. Laurent (2005) has recently found an alternative proof, using ideas
from real algebraic geometry.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 24 / 58
The Algebraic Variety of a TMP
Recall that if γ(2n) admits a rep. meas., then
M(n) ≡ M(n)(γ) ≥ 0
M(n) is RG (10.1)
rank M(n) ≤ card V(γ).
Question
Assume M(n) satisfies (10.1), and M(n) is singular. Does γ admit a rep.
meas.?
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 25 / 58
Question
For which p ∈ Pn do (10.1) and p(Z , Z ) = 0 imply that γ has a rep.
meas.?
(Fialkow) Consider p(z , z) ≡ zk − q(z), with deg q < k. If k is
minimal, if γ satisfies (10.1) and if p(Z , Z ) = 0, then
B := {1,Z ,Z 2, ...,Z k−1} is lin. indep. Moreover,
k ≥ card V(γ) ≥ rankM(n) ≥ k, so B is indeed a basis for CM(n).
It follows that M(n) is flat, and it therefore admits a k-atomic rep.
meas.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 26 / 58
The Quartic Moment Problem
Recall the lexicographic order on the rows and columns of M(2):
1,Z , Z ,Z 2, ZZ , Z 2
Z = A 1 (Dirac measure)
Z = A 1 + B Z (supp µ ⊆ line)
Z 2 = A 1 + B Z + C Z (flat extensions always exist)
ZZ = A 1 + B Z + C Z + D Z 2
D = 0 ⇒ ZZ = A 1 + B Z + B Z and C = B
⇒ (Z − B)(Z − B) = A + |B|2
⇒ WW = 1 (circle), for W :=Z − B√A + |B|2
.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 27 / 58
The functional calculus we have constructed is such that p(Z , Z ) = 0
implies supp µ ⊆ Z(p).
When{1,Z , Z ,Z 2, ZZ
}is a basis for CM(2), the associated algebraic
variety is the zero set of a real quadratic equation in
x := Re[z ] and y := Im[z ].
Using the flat data result, one can reduce the study to cases corresponding
to the following four real conics:
(a) W 2 = −2iW + 2iW −W 2 − 2WW parabola; y = x2
(b) W 2 = −4i1 + W 2 hyperbola; yx = 1
(c) W 2 = W 2 pair of intersect. lines; yx = 0
(d) WW = 1 unit circle; x2 + y2 = 1.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 28 / 58
Theorem QUARTIC
(RC-LF, 2005) Let γ(4) be given, and assume M(2) ≥ 0 and{1 ,Z , Z ,Z 2, ZZ
}is a basis for CM(2). Then γ(4) admits a rep. meas. µ.
Moreover, it is possible to find µ with card supp µ = rank M (2), except
in some cases when V(γ(4)) is a pair of intersecting lines, in which cases
there exist µ with card supp µ ≤ 6.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 29 / 58
Consider now the following property for a polynomial P ∈ Rn[x , y ]:
β ≡ β(2n) has a rep. meas. supported in Z(P) if and only if (A′n)
M(n)(β) is positive semi-definite, RG ,
P(X ,Y ) = 0 in CM(n), and rank M(n) ≤ card V(M(n)).
Polynomials which satisfy (A′n) form an attractive class, because if P
satisfies (A′n), then the degree-2n moment problem on P(x , y) = 0 can be
solved by concrete tests involving only elementary linear algebra and the
calculation of roots of polynomials.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 30 / 58
Theorem
(RC-LF, 2005) If deg P ≤ 2, then P satisfies (A′n) for every n ≥ deg P.
Despite this theorem, there are differences between the parabolic and
elliptic moment problems, and the hyperbolic problem. In the former
cases, the conditions of (A′n) always imply the existence of a
rank M(n)-atomic rep. meas., corresponding to a flat extension of
M(n); for this reason, positive Borel measures supported on these
curves always admit Gaussian cubature rules, i.e., rank M(n)-atomic
cubature rules of degree 2n (Fialkow and Petrovic, 2003). By
contrast, in the hyperbolic case, minimal rep. meas. µ sometimes
entail card supp µ > rank M(n) (and Gaussian cubature rules may
fail to exist).
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 31 / 58
The above Theorem is motivated in part by results of J. Stochel, who
solved the full moment problem on planar curves of degree at most 2.
Paraphrasing Stochel’s work (i.e., translating from the language of
moment sequences into the language of moment matrices), we
consider the following property of a polynomial P:
β(∞) has a rep. meas. supported in P(x , y) = 0 (A)
if and only if M(∞)(β) ≥ 0 and P(X ,Y ) = 0 in CM(∞).
Theorem
(Stochel, 1992) If deg P ≤ 2, then P satisfies (A).
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 32 / 58
Stochel also proved that there exist polynomials of degree 3 that do
not satisfy (A).
The link between TMP and FMP is provided by another result of
Stochel (2001):
Theorem
β(∞) has a rep. meas. supported in a closed set K ⊆ R2 if and only if,
for each n, β(2n) has a rep. meas. supported in K.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 33 / 58
Recall
Riesz-Haviland: There exists µ with supp µ ⊆ K ⇔ Λγ(p) ≥ 0 for all p
such that p|K ≥ 0.
For TMP, the natural analogue won’t work; for example, if
d = 1, K = R, and
M(2) :=
1 1 1
1 1 1
1 1 2
≥ 0,
then Λβ is R-positive, but no r.m. exists.
Possible Analogue: β(2n) admits a K -r.m. if and only if Λβ admits a
K -positive extension Λ : P2n+2 → R.
Another useful and relevant notion is that of V-positivity.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 34 / 58
Extremal MP
βi =
∫Rd
x i dµ, |i | ≤ 2n; (12.1)
P ≡ Rd [x ] = R[x1, ..., xd ] : space of real valued d-variable polynomials
Pk ≡ Rdk [x ] : the subspace of P consisting of polynomials p with
deg p ≤ k (k ≥ 1).
Λ ≡ Λβ : P2n → R (Riesz functional): if p(x) ≡∑
|i |≤2n aixi , then
Λ(p) :=∑
|i |≤2n aiβi
In the presence of a representing measure µ, we have Λ(p) =∫
p dµ.
p : coefficient vector (ai ) of p.
M(n) ≡M(n)(β) : moment matrix, with rows and columns X i indexed
by the monomials of Pn in degree-lexicographic order
d = n = 2 : the columns of M(2) are denoted as 1,X1,X2,X21 ,X2X1,X
22
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 35 / 58
M(n) is a real symmetric matrix characterized by
〈M(n)p, q〉 = Λ(pq) (p, q ∈ Pn). (12.2)
If µ is a representing measure for β, then
〈M(n)p, p〉 = Λ(p2) =∫
p2dµ ≥ 0; it follows that M(n) ≥ 0).
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 36 / 58
The algebraic variety of β is
V ≡ Vβ :=⋂
p∈Pn,p∈kerM(n)
Zp,
where Zp = {x ∈ Rd : p(x) = 0}.
If β admits a representing measure µ, then
p ∈ Pn satisfies p ∈ kerM(n) ⇔ supp µ ⊆ Zp
Thus supp µ ⊆ V, so r := rank M(n) and v := card V satisfy
r ≤ card supp µ ≤ v .
If p ∈ P2n and p|V ≡ 0, then Λ(p) =∫
p dµ = 0.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 37 / 58
Basic necessary conditions for the existence
of a representing measure
(Positivity) M(n) ≥ 0 (12.3)
(Consistency) p ∈ P2n, p|V ≡ 0 =⇒ Λ(p) = 0 (12.4)
(Variety Condition) r ≤ v , i.e., rank M(n) ≤ card V. (12.5)
Consistency implies
(Recursiveness) p, q, pq ∈ Pn, p ∈ kerM(n) =⇒ pq ∈ kerM(n). (12.6)
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 38 / 58
Previous results:
For d = 1 (the T Hamburger MP for R), positivity and recursiveness
are sufficient
For d = 2, there exists M(3) > 0 for which β has no representing
measure
In general, Positivity, Consistency and the Variety Condition are not
sufficient.
Question C
Suppose M(n)(β) is singular. If M(n) is positive, β is consistent, and
r ≤ v, does β admit a representing measure?
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 39 / 58
More generally, the following question remained unsolved until very
recently.
Question RG
Suppose M(n)(β) is singular. If M(n) is positive, recursively generated,
and r ≤ v, does β admit a representing measure?
RC-LF: If d = 2 and M(n)p = 0 for some p with deg p ≤ 2, then
Question RG has an affirmative answer. (Theorem Quartic)
RC-LF-MM: If d = 2, y − x3 ∈ kerM(n) and r = v ≤ 7, then
Question RG has an affirmative answer.
RC-LF-MM: If d = 2, y − x3 ∈ kerM(n) and r = v = 8, then
Question RG has a negative answer.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 40 / 58
The next result gives an affirmative answer to Question C in the extremal
case, i.e., r = v .
Theorem EXT
(RC-LF and M. Moller, 2005) For β ≡ β(2n) extremal, i.e., r = v, the
following are equivalent:
(i) β has a representing measure;
(ii) β has a unique representing measure, which is rank M(n)-atomic
(minimal);
(iii) M(n) ≥ 0 and β is consistent.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 41 / 58
In many cases, the conditions of Theorem EXT provide a concrete
solution to the extremal case of TMP. Indeed, only elementary linear
algebra is required to verify that M(n) ≥ 0, to compute rank M(n),
and to identify the column relations which define V.
If the points of V can be computed exactly, then only elementary
linear algebra is required to verify that β is consistent.
Question RG is significant because recursiveness is generally a simpler
condition to work with than consistency. For example, we often have
M(n) ≥ 0 and M(n − 1) > 0 (positive definite), in which case M(n)
is obviously recursively generated, but we do not know whether r ≤ v
implies that M(n) is consistent in this case.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 42 / 58
The extremal case is inherent in TMP:
C. Bayer and J. Teichmann (2006) (extending a classical theorem of
V. Tchakaloff and its successive generalizations by I.P. Mysovskikh,
M. Putinar and RC-LF) recently proved that if β(2n) has a
representing measure, then it has a finitely atomic representing
measure;
RC-LF showed that β(2n) has a finitely atomic representing measure if
and only if M(n) admits an extension to a positive moment matrix
M(n + k) (for some k ≥ 0, which in turn admits a rank-preserving
(i.e., flat) moment matrix extension M(n + k + 1);
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 43 / 58
in many instances, M(n + k + 1) is an extremal moment matrix for
which there is a computable rankM(n + k)-atomic representing
measure µ. Clearly, µ is also a finitely atomic representing measure
for β(2n), and every finitely atomic representing measure for β(2n)
arises in this way.
there exist extremal TCMP of arbitrarily large degree
Moment theory can sometimes be used to estimate the number and
location of the zeros of a prescribed polynomial; for example, we can
use TMP tecniques to show that the polynomial
p(z)≡ z2n + az2n−1 − az − 1 (0 < a < 1)
has 2n distinct zeros, all in the unit circle.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 44 / 58
Real Ideals and Necessary Conditions
If β(2n) has a representing measure, then the Riesz functional
Λ : P2n → R, Λ(x i ) := βi ≡∫
Rd
x i dµ (|i | ≤ 2n),
is square positive, that is,
p ∈ Pn ⇒ Λ(p2) ≥ 0.
If we assume that for a representing measure µ all moments∫Rd
x idµ, i ∈ Zd+
are convergent, then we can extend Λ to P by letting
Λ(x i ) :=
∫Rd
x idµ, i ∈ Zd+,
thus obtaining a square positive functional over P.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 45 / 58
Under this assumption the set
I := {p ∈ P : Λ(p2) = 0}
is a real ideal, i.e., it is an ideal ( p1, p2 ∈ I ⇒ p1 + p2 ∈ I and
p ∈ I, q ∈ P ⇒ pq ∈ I) and satisfies one of the two equivalent
conditions:
(i) For s ∈ Z+, p1, . . . , ps ∈ P :∑s
i=1 p2i ∈ I ⇒ {p1, . . . , ps} ⊆ I
(ii) There exists G ⊆ Rdsuch that for all p ∈ P : p|G ≡ 0 ⇒ p ∈ I.
If I is a real ideal, then one may take for G ⊆ Rd the real variety
VR(I) := {w ∈ Rd : f (w) = 0 (all f ∈ I)}.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 46 / 58
But one may also take any subset G of VR(I) containing sufficiently many
points, such that
p ∈ P, p|G ≡ 0 ⇒ p|VR(I) ≡ 0.
For instance, if the real variety is a (real) line, one may take for G a subset
of infinitely many points on that line. On the other hand, if VR(I) is a
finite set of points, then necessarily G = VR(I).
If I is an ideal, its subset Ik := I ∩ Pk is an R-vector subspace of Pk .
One can then introduce the Hilbert function of I by
HI(k) := dimPk − dim Ik , k ∈ Z+.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 47 / 58
Both k 7→ dim Ik and k 7→ HI(k) are nondecreasing functions.
For sufficiently large k, say k ≥ k0, HI(k) becomes a polynomial in k,
the so called Hilbert polynomial of I, whose degree equals the
dimension of I.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 48 / 58
Assume now that β(2n) admits a representing measure µ. Then,
irrespective of whether the Riesz functional Λ : Pn → R can be extended
to a square positive functional Λ : P → R, we can define the ideal
I(µ) := {p ∈ P : p|supp µ ≡ 0}. (13.1)
Since supp µ ⊆ Rd , I(µ) is a real ideal, which we will call the real ideal of
β(2n).
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 49 / 58
Lemma
Assume β(2n) has a representing measure, and let I(µ) be its real ideal.
Then
{p ∈ Pn : M(n)p = 0} = I(µ)⋂Pn. (13.2)
If t1, ..., tN denote the monomials x i ∈ Pn in degree-lexicographic order,
then the row vectors of M(n) and the row vectors of
Wn := {(t1(w), ..., tN(w)) : w ∈ supp µ} span the same subspace of RN ;
in particular, rank M(n) = HI(µ)(n).
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 50 / 58
Lemma
Let Λ : P2n → R be a linear functional and let V ≡ {w1, ...,wm} ⊆ Rd .
The following statements are equivalent.
(a) There exist αi ∈ R such that Λ(p) =∑m
i=1 αip(wi ) (all p ∈ P2n).
(b) (Consistency) If p ∈ P2n and p|V ≡ 0, then Λ(p) = 0.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 51 / 58
Moment Matrices and Consistency
Recall that
V ≡ Vβ :=⋂
p∈Pn,p(X )=0
Zp.
Let Pn|V denote the restriction to V of the polynomials in Pn, and
consider the mapping ϕβ : CM(n) → Pn|V given by p(X ) 7→ p|V . The
map ϕβ is well-defined, and β has a representing measure µ, then ϕβ
is 1-1.
Proposition
Let β, ϕβ and M(n)(β) be as before. Then
β consistent =⇒ ϕβ 1-1 =⇒ M(n)(β) recursively generated.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 52 / 58
Proposition
For d = 2 (the plane), if M(n)(β) is recursively generated and Vβ is a
proper, infinite irreducible curve, then β is consistent.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 53 / 58
Solution of the Extremal Moment Problem
Assume that β ≡ β(2n) is extremal, i.e., r := rank M(n) and v := card Vsatisfy r = v . Let V ≡ {w1, ...,wr} denote the distinct points of V. If µ
is a representing measure for β, then since supp µ ⊆ V and
r ≤ card supp µ ≤ v , the extremal hypothesis r = v implies that
supp µ = V. Thus µ is necessarily is of the form
µ =r∑
i=1
ρiδwi . (15.1)
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 54 / 58
Let p1, ..., pr be polynomials in Pn such that B ≡ {p1(X ), ..., pr (X )} is a
basis for the column space of M(n), and set
W ≡ WB :=
p1(w1) . . . p1(wr )
. . . . .
. . . . .
. . . . .
pr (w1) . . . pr (wr )
.
Also, recall that if Pn|V denote the restriction to V of the polynomials in
Pn, the map ϕβ : CM(n) → Pn|V is given by p(X ) 7→ p|V .
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 55 / 58
Lemma
The following are equivalent for β extremal:
i) ϕB is 1-1, i.e., p ∈ Pn, p|V ≡ 0 =⇒ p(X ) = 0 in CM(n);
ii) For any basis B of CM(n), WB is invertible;
iii) There exists a basis B for CM(n) such that WB is invertible.
Theorem
(RC-LF and M. Moller, 2005) For β ≡ β(2n) extremal, TFAE:
(i) β has a representing measure;
(ii) β has a unique representing measure, which is a rank M(n)-atomic;
(iii) For some (respectively, for every) basis B for CM(n), WB is invertible
and µB is a representing measure for β;
(iv) β is consistent and M(n) ≥ 0.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 56 / 58
An Example with r < v < +∞
Example
Consider
M(3) =
1 0 0 1 2 5 0 0 0 0
0 1 2 0 0 0 2 5 14 42
0 2 5 0 0 0 5 14 42 132
1 0 0 2 5 14 0 0 0 0
2 0 0 5 14 42 0 0 0 0
5 0 0 14 42 132 0 0 0 0
0 2 5 0 0 0 5 14 42 132
0 5 14 0 0 0 14 42 132 429
0 14 42 0 0 0 42 132 429 2000
0 42 132 0 0 0 132 429 2000 338881
.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 57 / 58
We have M(3) ≥ 0, M(2) > 0, r = 8,
Y = X 3, (16.1)
and
Y 3 = q(X ,Y ). (16.2)
where q(x , y) := −2285x + 5720y − 34441yx2 + 578y2x . Here v = 9.
Thus, M(3) is positive, recursively generated, r < v , and the minimal
representing measure for β(6) is v -atomic.
Raul E. Curto, University of Iowa (Iowa) Extremal TMP 58 / 58