Multivariable Matrix-valued moment problems David P. Kimsey The - - PowerPoint PPT Presentation

Multivariable Matrix-valued moment problems

David P. Kimsey The Weizmann Institute of Science Operator theory and operator algebras 2012 May 22, 2012 Partially supported by NSF grant DMS-0901628

The classical moment problem

Full moment problem: (i) Given tsm✉m P N0 we wish to determine whether or not there exists σ P m♣Rq such that sm ✏ ➺

R

xmdσ♣xq, m P N0 :✏ t0, 1, . . .✉. (1) (ii) Describe all σ which obey (1). The cases when supp σ ❸ r0, ✽q, R, and rc, ds are due to Stieltjes, Hamburger, and Hausdorff, respectively. Observation: If σ satisfies (1) then

n

➳

a,b✏0

za¯ zbsab ✏

n

➳

a,b✏0

za¯ zb ➺

R

xabdσ♣xq ✏ ➺

R

✞ ✞ ✞ ✞ ✞

n

➳

a✏0

zaxa ✞ ✞ ✞ ✞ ✞

2

dσ♣xq ➙ 0.

Hamburger’s theorem

Theorem (Hamburger’s theorem, 1921): tsm✉m P N0 has a representing measure if and only if

n

➳

a,b✏0

za¯ zbsab ➙ 0 ð ñ ♣sabqn

a,b✏0 :✏

☎ ✝ ✆ s0 ☎ ☎ ☎ sn . . . ... . . . sn ☎ ☎ ☎ s2n ☞ ✍ ✌➞ 0, for all finite subsets tz1, . . . , zn✉ ⑨ C. Remarks on proofs of Hamburger’s theorem:

➓ Hamburger’s original proof is around 150 pages. ➓ The operator theory proof is much shorter. [Krein, 1949]

generalized this result to the case when one is a given tSm✉m P N0.

The full multidimensional K-moment problem

Full K-moment problem on Rd: Given tsm✉m P Nd

0 and K ❸ Rd, we

wish to determine whether or not there exists σ such that sm ✏ ➺

Rd xmdσ♣xq :✏

➺ ☎ ☎ ☎ ➺

Rd xm1 1

☎ ☎ ☎ xmd

d dσ♣x1, . . . , xdq, m P Nd 0.

(2) and supp σ ❸ K. (3) Observation: If tsm✉m P Nd

0 has a representing measure then

➳

0↕⑤m⑤,⑤ r m⑤↕n

zm¯ z r

msm r m ➙ 0, n P N0.

(4) Caveat: There do exist sequences tsm✉m P Nd

0 which satisfy (4) yet

do not satisfy (2).

Solutions to the full multidimensional K-moment problem

➓ A solution to the full K-moment problem on R in [M. Riesz,

1923].

➓ A d-variable (d → 1) generalization was achieved in [Haviland,

1936].

➓ When K is compact and semi-algebraic, [Schm¨

udgen, 1991] has a solution to the full K-moment problem based on the approach in [Haviland, 1936].

➓ Subsequently, [Putinar & Vasilescu, 1999] improved upon this

approach.

Truncated Hamburger moment problem

Problem: Given a real-valued sequence tsk✉0↕k↕n, we wish to determine necessary and sufficient conditions on tsk✉0↕k↕n so that there exists σ so that sk ✏ ➺

R

xkdσ♣xq, 0 ↕ k ↕ n. Remarks: When n ✏ 2m then a solution exists if and only if we can choose s2m1 and s2m2 such that ☎ ✝ ✝ ✝ ✝ ✝ ✆ s0 ☎ ☎ ☎ sm✁1 sm sm1 . . . ... . . . . . . . . . sm✁1 ☎ ☎ ☎ s2m✁2 s2m✁1 s2m sm ☎ ☎ ☎ s2m✁1 s2m s2m1 sm1 ☎ ☎ ☎ s2m s2m1 s2m2 ☞ ✍ ✍ ✍ ✍ ✍ ✌ ➞ 0. In this case, one can find σ ✏ ➦r

j✏1 ρjδxj, where

r ✏ rank

• sij

✟m

i,j✏0.

Multivariable THMPs

Truncated Hamburger moment problem (even total degree): Given tsm✉0↕⑤m⑤↕2n we wish to determine whether or not there exists σ such that sm ✏ ➺

Rd xmdσ♣xq :✏

➺ ☎ ☎ ☎ ➺

Rd x1m1 ☎ ☎ ☎ xd mddσ♣x1, . . . , xdq.

Given tsm✉0↕⑤m⑤↕2n we can construct the following moment matrix: M♣nq :✏

• sm r

m

✟

0↕⑤m⑤,⑤ r m⑤↕n .

When d ✏ 2 and n ✏ 1, M♣1q :✏ ☎ ✆ s00 s01 s10 s01 s02 s11 s10 s11 s20 ☞ ✌

Flat extension theory of Curto and Fialkow

Given tsm✉0↕⑤m⑤↕2n, we call M♣n 1q a flat extension of M♣nq when there exist new data ts r

m✉2n1↕⑤ r m⑤↕2n2 such that

• 1. M♣n 1q ✏ ♣sm r

mq0↕⑤m⑤,⑤ r m⑤↕n1 ➞ 0

• 2. rank M♣n 1q ✏ rank M♣nq.

For instance, when we are given ts00, s01, s10, s02, s11s20✉ so that M♣1q ➞ 0, we wish to find ts03, s12, s21, s30, s04, s13, s22, s31, s40✉ so that

M♣2q :✏ ☎ ✝ ✝ ✝ ✝ ✝ ✝ ✆ s00 s01 s10 s02 s11 s20 s01 s02 s11 s03 s12 s21 s10 s11 s20 s12 s21 s30 s02 s03 s12 s04 s13 s22 s11 s12 s21 s13 s22 s31 s20 s21 s30 s22 s31 s40 ☞ ✍ ✍ ✍ ✍ ✍ ✍ ✌ ➞ 0 and rank M♣1q ✏ rank M♣2q.

Theorem (Curto and Fialkow, 1996): The even total degree HMP has a solution if and only if M♣nq eventually admits a flat extension.

Truncated matrix-valued Hamburger moment problem

Problem: Given tSm✉0↕m↕n, we wish to find Σ :✏ ♣σjkqp

j,k✏1 such

that Sm ✏ ➺

R

xmdΣ♣xq :✏ ➩

R xmdσjk♣xq

✟p

j,k✏1 , 0 ↕ m ↕ n.

This problem has been studied by

➓ [Dym, 1989]; ➓ [Bolotnikov, 1996]; ➓ [Bakonyi & Woerdeman, 2011].

Under analogous conditions, one can find σ ✏ ➦r

j✏1 Tjδxj, where

➦r

j✏1 rank Tj ✏ rank

• Sij

✟n

i,j✏0.

An illustrative example

Given the sequence ts♣k,ℓq✉0↕kℓ↕3, suppose Φ ✏ ☎ ✆ s00 s01 s10 s01 s02 s11 s10 s11 s20 ☞ ✌✏ ☎ ✆ 3 1 2 1 1 1 2 1 2 ☞ ✌→ 0, Φ1 ✏ ☎ ✆ s10 s11 s20 s11 s12 s21 s20 s21 s30 ☞ ✌✏ ☎ ✆ 2 1 2 1 1 1 2 1 2 ☞ ✌, and Φ2 ✏ ☎ ✆ s01 s02 s11 s02 s03 s12 s11 s12 s21 ☞ ✌✏ ☎ ✆ 1 1 1 1 1 1 1 1 1 ☞ ✌.

An illustrative example continued

Realize that Θ1 ✏ Φ✁1Φ1 ✏ ☎ ✆ 1 1 1 ☞ ✌ and Θ2 ✏ Φ✁1Φ2 ✏ ☎ ✆ 1 1 1 ☞ ✌ commute. Also the eigenvalues of Θ1 and Θ2 are t0, 1, 1✉ and t0, 0, 1✉, respectively. Put ♣x1, y1q ✏ ♣0, 0q, ♣x2, y2q ✏ ♣1, 0q and ♣x3, y3q ✏ ♣1, 1q. Then σ ✏ ➦3

j✏1 δ♣xj,yjq is a solution!

Indexing sets

➓ We say a finite set Γ ⑨ Nd 0 is a lattice set when for all γ P Γ

there exist γ1 ✏ 0d, γ2, . . . , γk P Γ and j1, . . . , jk P t1, . . . , d✉ so that γ2 ✏ γ ej1, . . . , γ ✏ γk ejk, where k ✏ ⑤γ⑤.

➓ We say a finite set Γ ⑨ Nd 0 is lower inclusive when for any

m ✏ ♣m1, . . . , mdq P Nd

0 and γ ✏ ♣g1, . . . , gdq P Γ with

mj ↕ gj, 1 ↕ j ↕ d, we have that m P Γ.

➓ Note that t♣0, 0q, ♣0, 1q, ♣1, 1q✉ ⑨ N2 0 is a lattice set but not a

lower inclusive set. Also t♣0, 0q, ♣1, 1q✉ ⑨ N2

0 is not a lattice

set.

Truncated matrix-valued K-moment problem

• n Rd

Problem: Let K ❸ Rd, Γ ⑨ Nd

0 be a lattice set and tSγ✉γPΓ be

• given. We wish to determine whether or not there exists Σ so that

(i) Sγ ✏ ➩

Rd xγdΣ♣xq, for all γ P Γ;

(ii) supp Σ ❸ K.

➓ When Γ ✏ tm P Nd

0 : 0 ↕ ⑤m⑤ ↕ 2n✉ and tSγ✉γPΓ is scalar-valued then

[Curto and Fialkow, 2000 & 2005] analyzed the K-moment problem on Rd and its multidimensional complex analogue.

➓ [Stochel, 2001] showed that the full K-moment problem on Rd or Cd has

a solution if and only if the truncated K-moment problem, where the moments given have indices which are of total degree at most 2n, has a solution for every n P N.

Construction of moment matrices from given data

➓ Given a lattice set Λ ⑨ Nd 0, we define

ΛΛ ✏ tλr λ: λ, r λ P Λ✉ and ΛΛej ✏ tλr λej : λ, r λ P Λ✉, 1 ↕ j ↕ d.

➓ Put Γ ✏ ♣Λ Λq ❨ ♣Λ Λ e1q ❨ ☎ ☎ ☎ ❨ ♣Λ Λ edq,

which will serve as the indexing set for tSγ✉γPΓ.

➓ Introduce Φ, Φ1, . . . , Φd as follows. Index the rows and

columns of Φ by Λ. Let the entry in the row indexed by λ and the column indexed by r λ be given by Sλr

λ. That is,

Φ ✏ ♣Sλr

λqλ, r λ P Λ ➞ 0. ➓ Similarly, index the rows and columns of Φj by Λ. That is,

Φj ✏ ♣Sλr

λejqλ, r λ P Λ, 1 ↕ j ↕ d.

An example of the construction

Let tSγ✉γPΓ, where Γ ✏ tγ P Nd

0 : 0 ↕ ⑤γ⑤ ↕ 3✉. Put

Λ ✏ t♣0, 0q, ♣0, 1q, ♣1, 0q✉ and so then we get the following matrices: Φ ✏ ♣Sλ˜

λqλ,˜ λPΛ ✏

☎ ✆ S00 S01 S10 S01 S02 S11 S10 S11 S20 ☞ ✌➞ 0, Φ1 ✏ ♣Sλ˜

λe1qλ,˜ λPΛ ✏

☎ ✆ S10 S11 S20 S11 S12 S21 S20 S21 S30 ☞ ✌, and Φ2 ✏ ♣Sλ˜

λe2qλ,˜ λPΛ ✏

☎ ✆ S01 S02 S11 S02 S03 S12 S11 S12 S21 ☞ ✌.

Minimal representing measures

Given tSγ✉γPΓ, where Γ ✏ Λ Λ, build Φ ✏ ♣Sλµqλ,µPΛ. Suppose tSγ✉γPΓ has a representing measure Σ ✏ ➦k

q✏1 Tqδwq, where

T1, . . . , Tk ➞ 0 and w1, . . . , wk are distinct points in Rd. We say that Σ is minimal when rank Φ ✏ ➦k

q✏1 rank Tq.

Why? For m P Nd

0, realize Sm :✏

➩ ξmdσ♣ξq ✏ ➦k

q✏1 Tqwm q . One

can check that Φ ✏ ♣V ❜ IpqT♣T1 ❵ ☎ ☎ ☎ ❵ Tkq♣V ❜ Ipq, where V :✏ ☎ ✝ ✆ wλ1

1

☎ ☎ ☎ wλk

1

. . . . . . wλ1

k

☎ ☎ ☎ wλk

k

☞ ✍ ✌.

K-inclusive eigenvalue property

Let K ❸ Rd and W1, . . . , Wn P Cn✂n. We say W1, . . . , Wd has the K-inclusive eigenvalue property with respect to the subspace M ❸ Cn, with dim M ✏ k, if the following conditions are satisfied:

• 1. W ✝

j is M-invariant, i.e.

Wj ✏ ✂⑨ Wj ✝ ✝ ✡ : M M ❵ Ñ ❵ M❑ M❑ , 1 ↕ j ↕ d;

• 2. There exists an invertible matrix S so that

S✁1⑨ W1S ✏ diag ✁ x♣1q

1 , . . . , x♣kq 1

✠ ; . . . S✁1⑨ WdS ✏ diag ✁ x♣1q

d , . . . , x♣kq d

✠ . 3. ✁ x♣qq

1 , . . . , x♣qq d

✠ P K, 1 ↕ q ↕ k.

Minimal truncated matrix-valued K-moment problem on Rd solution

Theorem (Kimsey & Woerdeman, 2010): Let K ❸ Rd and tSγ✉γPΓ, where Γ ✏ ♣Λ Λq ❨ ♣Λ Λ e1q ❨ . . . ❨ ♣Λ Λ edq, be given. There exists a minimal K-representing measure Σ if: (i) Φ ➞ 0; (ii) There exist Θ1, . . . , Θd so that

(a) ΦΘj ✏ Φj, 1 ↕ j ↕ d; (b) Θ1, . . . , Θd have the K-inclusive eigenvalue property with respect to M ✏ Ran Φ.

Conversely, if Σ ✏ ➦k

q✏1 Tqδwq is given then there exists a lower

inclusive set Λ which gives rise to Φ, Φ1, . . . , Φd which satisfy conditions (i) and (ii).

Remarks on the proof

➓ To prove the sufficiency of conditions (i) and (ii), we use the

fact that ⑨ W1, . . . , ⑨ Wd commute to write Φ ✏ CC ✝ and Φj ✏ CDjC ✝, 1 ↕ j ↕ d, where C is an injective matrix and D1, . . . , Dd are real diagonal matrices.

➓ Next, write C ✏ col♣CλqλPΛ and it turns out that

Sγ ✏ C0dDg1

1 ☎ ☎ ☎ Dgd d C ✝ 0d,

for all γ ✏ ♣g1, . . . , gdq P Γ.

Remarks on the proof continued

➓ To prove the necessity of conditions (i) and (ii), use a

construction in [Sauer, 1997] to produce a lower inclusive set ˜ Λ ✏ tλ1, . . . λk✉ ⑨ Nd

0 so that

V :✏ ☎ ✝ ✆ wλ1

1

☎ ☎ ☎ wλk

1

. . . . . . wλ1

k

☎ ☎ ☎ wλk

k

☞ ✍ ✌ is invertible.

➓ Next put Sm ✏

➩ xmdΣ♣xq and note that Sm ✏ ➦k

q✏1 Tqwm q .

Write wq ✏ ♣w♣qq

1

, . . . , w♣qq

d q P K, 1 ↕ q ↕ k. So then we get

Φ ✏ ♣V ❜ IpqTR♣V ❜ Ipq ➞ 0 and Φj ✏ ♣V ❜ IpqTRXj♣V ❜ Ipq, where R ✏ T1 ❵ ☎ ☎ ☎ ❵ Tk and Xj ✏ x♣1q

j

Ip ❵ ☎ ☎ ☎ ❵ x♣kq

j

Ip, 1 ↕ j ↕ d. Put Θj ✏ ♣V ❜ Ipq✁1Xj♣V ❜ Ipq so that condition (ii) is satisfied.

Truncated matrix-valued K-moment problem

• n Cd

The power moments of Σ are defined by the formula ˆ Σ♣m, ˜ mq ✏ ➺

Cd ¯

zmz ˜

mdΣ♣zq :✏

➺ ☎ ☎ ☎ ➺

Cd ¯

zm1

1

☎ ☎ ☎ ¯ zmd

d z ˜ m1 1

☎ ☎ ☎ z ˜

md d ,

where m, ˜ m P Nd

0.

Note that ˆ Σ♣m, ˜ mq✝ ✏ ˆ Σ♣ ˜ m, mq. Problem: Given a lattice set Γ ⑨ Nd

0 ✂ Nd 0 and K ❸ Cd, and

tS♣γ,˜

γq✉♣γ,˜ γqPΓ, we look for Σ so that

(i) ˆ Σ♣γ, ˜ γq ✏ S♣γ,˜

γq, for all ♣γ, ˜

γq P Γ; (ii) supp Σ ❸ K.

Indexing sets and moment matrices

➓ Given a lattice set Λ ⑨ Nd 0 ✂ Nd 0, we define the set

ΛT ✏ t♣µ, λq: ♣λ, µq P Λ✉. For example, if Λ ✏ t♣0, 0q, ♣0, 1q, ♣1, 0q, ♣2, 0q✉ then ΛT ✏ t♣0, 0q, ♣0, 1q, ♣0, 2q♣1, 0q✉.

➓ Put

Γ ✏ ♣Λ ΛTq ❨

d

↕

j✏1

♣Λ ΛT ♣0d, ejqq ❨ ♣Λ ΛT ♣ej, 0dqq, which will serve as an indexing set for tS♣γ,˜

γq✉♣γ,˜ γqPΓ. ➓ Index the rows of Φ by Λ and the columns by ΛT. Let the

entry in the row indexed by ♣α, βq P Λ and the column indexed by ♣µ, λq P ΛT be given by S♣α,βq♣µ,λq. That is, Φ ✏ ♣S♣αµ,βλqq♣α,βqPΛ, ♣µ,λqPΛT .

Moment matrix construction continued

➓ Index the rows of Φzj by Λ and the columns by ΛT. Let the

entry in the row indexed by ♣α, βq P Λ and the column indexed by ♣µ, λq P ΛT be given by S♣α,βq♣µ,λq♣0d,ejq. That is, Φzj ✏ ♣S♣α,βq♣µ,λq♣0d,ejqq♣α,βqPΛ, ♣µ,λqPΛT , 1 ↕ j ↕ d.

➓ Index the rows of Φ¯ zj by Λ and the columns by ΛT. Let the

entry in the row indexed by ♣α, βq P Λ and the column indexed by ♣µ, λq P ΛT be given by S♣α,βq♣µ,λq♣ej,0dq. That is, Φ¯

zj ✏ ♣S♣α,βq♣µ,λq♣ej,0dqq♣α,βqPΛ, ♣µ,λqPΛT , 1 ↕ j ↕ d.

Note that we necessarily have Φ ✏ Φ✝ because if tS♣γ,˜

γq✉♣γ,˜ γqPΓ

has a representing measure then S♣γ,˜

γq ✏ S✝ ♣˜ γ,γq, for all ♣γ, ˜

γq P Γ. Similarly, we have Φ✝

zj ✏ Φ¯ zj.

An example of the construction

Let Λ ✏ t♣0, 0q, ♣0, 1q, ♣1, 0q✉. Then ΛT ✏ t♣0, 0q, ♣1, 0q, ♣0, 1q✉ and Γ ✏ ♣Λ ΛTq ❨ ♣Λ ΛT ♣1, 0qq ❨ ♣Λ ΛT ♣0, 1qq ✏ t♣m, ˜ mq P N2

0 : 0 ↕ m ˜

m ↕ 3✉. We get the following matrices: Φ ✏ ☎ ✆ S00 S10 S01 S01 S11 S02 S10 S20 S11 ☞ ✌➞ 0, Φ¯

z ✏

☎ ✆ S10 S20 S11 S11 S21 S12 S20 S30 S21 ☞ ✌, and Φz ✏ ☎ ✆ S01 S11 S02 S02 S12 S12 S11 S21 S12 ☞ ✌.

Matrix-valued truncated K-moment problem

• n Cd

Theorem (Kimsey & Woerdeman, 2010): Let K ❸ Cd, Λ ⑨ Nd

0 ✂ Nd 0 be a lattice set, and suppose the Cp✂p-valued

sequence tS♣γ,˜

γq✉♣γ,˜ γqPΓT ✂Γ is given.

If

• 1. Φ ➙ 0;
• 2. There exist matrices Θz1, . . . , Θzd, Θ¯

z1, . . . , Θ¯ zd which

commute with respect to M ✏ Ran Φ so that ΦΘzj ✏ Φzj and ΦΘ¯

zj ✏ Φ¯ zj, 1 ↕ j ↕ d. In addition, Θz1, . . . , Θzd must

satisfy the K-inclusive eigenvalue property with respect to M ✏ Ran Φ. then there exists a minimal Σ such that S♣γ,˜

γq ✏

➩

Cd ¯

zγz ˜

γdΣ♣zq

and supp Σ ❸ K.

Remarks on a converse result

We only have a converse when d ✏ 1. The following question would lead to a d → 1 generalization. Question: Given distinct u1, . . . , uk P Cd. Can we choose a lattice set Λ ✏ t♣λ1, µ1q, . . . , ♣λk, µkq✉ P Nd

0 ✂ Nd 0 (lower inclusive set) so

that V ✏ ☎ ✝ ✆ ¯ uλ1

1 uµ1 1

☎ ☎ ☎ ¯ uλk

1 uµk 1

. . . . . . ¯ uλ1

k uµ1 k

☎ ☎ ☎ ¯ uλk

k uµk k

☞ ✍ ✌ is invertible?

Cubic complex moment problem

Problem: Given s :✏ ts♣m1,m2q✉0↕m1m2↕3 we wish to find σ so that s♣m, ˜

mq ✏

➺

C

¯ zmz ˜

mdσ♣zq.

Let

Φ ✏ ☎ ✆ s00 s10 s01 s01 s11 s02 s10 s20 s11 ☞ ✌, Φ¯

z ✏

☎ ✆ s10 s20 s11 s11 s21 s12 s20 s30 s21 ☞ ✌, and Φz ✏ ☎ ✆ s01 s11 s02 s02 s12 s12 s11 s21 s12 ☞ ✌.

Theorem (Kimsey, 2011): Suppose r :✏ rank Φ ✏ 1, 2. Then the following are equivalent: (i) s has a representing measure; (ii) s has a r-atomic representing measure; (iii) Φ ➞ 0 and there exist commuting Θz, Θ¯

z such that

ΦΘz ✏ Φz and ΦΘ¯

z ✏ Φ¯ z.

Suppose r ✏ 3. Then s has a 3-atomic representing measure if and

• nly if Φ✁1Φz and Φ✁1Φ¯

z commute.

Data that does not admit a minimal representing measure

Example: There does exist a sequence s with a representing measure, with rank Φ ✏ 3, yet s does not have a 3-atomic representing measure. Let s be given by

Φ ✏ ☎ ✆ 4 54 ✁50 4i ✁50 ✁ 4i 54 ☞ ✌, Φz ✏ ☎ ✆ ✁50 ✁ 4i 54 54 ✁50 ✁ 4i ☞ ✌, and Φ¯

z ✏

☎ ✆ 54 ✁ 4i ✁50 4i ✁50 4i 54 ☞ ✌. Hence Θz ✏ Φ✁1Φz ✏ ☎ ✆ ✁ 25

2 ✁ i 27 2

1 ☞ ✌and Θ¯

z ✏ Φ✁1Φ¯ z ✏

☎ ✆

27 2

✁ 25

2 i 25 2 i

✁ 27

2

☞ ✌which do not commute.

Note that a 4-atomic representing measure for s is given by σ ✏ δ5i δ1✁i δ✁5i δ✁1i.

Example continued

Suppose there does exist a 3-atomic representing measure for s. Then we must have the existence of s22 → 0, s23, s32, s13, s31 P C so that r Φ ✏ ☎ ✆ s00 s01 s02 s10 s11 s12 s20 s21 s22 ☞ ✌✏ ☎ ✆ 4 ✁50 ✁ 4i 54 ✁50 4i s22 ☞ ✌→ 0, r Φz ✏ ☎ ✆ s01 s02 s03 s11 s12 s13 s21 s22 s23 ☞ ✌✏ ☎ ✆ ✁50 ✁ 4i 54 s13 s22 s23 ☞ ✌, and r Φ¯

z ✏

☎ ✆ s10 s11 s12 s20 s21 s22 s30 s31 s32 ☞ ✌✏ ☎ ✆ 54 ✁50 4i s22 s31 s32 ☞ ✌ admit commuting r Θz, r Θ¯

z.

Tchakaloff’s theorem

Theorem (Kimsey, 2011): Suppose tSm✉0↕⑤m⑤↕n is a truncated sequence with representing measure Σ P M♣Rdq. There exists a finitely atomic r Σ P M♣Rdq, such that r Σ is a representing measure for tSλ✉0↕⑤m⑤↕n and supp r Σ ❸ supp Σ. Remarks:

➓ [Tchakaloff, 1957] proved a scalar-valued result when the

representing measure is absolutely continuous with respect to the Lebesgue measure.

➓ [Putinar, 1997] proved a scalar-valued result when the

representing measure has compact support.

➓ [Bayer & Teichmann, 2006] proved a scalar-valued result with

no assumptions on the representing measure.

➓ [Laurent, 2006] provided an alternative proof of Bayer &

Teichmann’s result which is more algebraic.

Remarks on the proof on Tchakaloff’s theorem

The basic idea is to build a convex cone C from matrices of the form col♣xmq0↕⑤m⑤↕n ❜ vv✝ where x P supp Σ and v P Cp. Next we have to show that

• Sm

✟

0↕⑤m⑤↕n P ri C.

Finaly use Carath´ eodory’s theorem to produce the finitely atomic representing measure.