Truncated Moment Problems: The Extremal Casehomepage.math.uiowa.edu/~rcurto/IWOTA2006Slides.pdf ·...

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.


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


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.


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.


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 .


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 .


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 .


Corollary

Let A ≥ 0 and assume rank

(A B

B∗ C

)=rank A. Then(

A B

B∗ C

)≥ 0.


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, . . .


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.


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

.


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.


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+


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


(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)


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.


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).


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


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 .


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.


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.


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.


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.


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.?


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.


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

.


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.


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.


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.


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).


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).


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.


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.


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


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).


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.


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)


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?


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.


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.


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.


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);


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.


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.


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)}.


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+.


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.


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).


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).


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.


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.


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.


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)


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 .


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.


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

.


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.


Truncated Moment Problems: The Extremal Casehomepage.math.uiowa.edu/~rcurto/IWOTA2006Slides.pdf ·...

Documents

Transcript of Truncated Moment Problems: The Extremal Casehomepage.math.uiowa.edu/~rcurto/IWOTA2006Slides.pdf ·...