Truncated Moment Problems with Associated Finite Algebraic Varieties - PowerPoint PPT Presentation

Truncated Moment Problems with Associated Finite Algebraic Varieties (joint work with Seonguk Yoo) Ra´ ul Curto Helton Workshop, UCSD, October 4, 2010 Dedicated to Bill on the occasion of his 65th birthday! Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 1 / 57

Outline of the Talk Brief Review of Full Moment Problem Truncated Moment Problems (Basic Positivity, Functional Calculus, Algebraic Variety) Moment Matrix Extension Approach Positive Linear Functional Approach TMP Version of the Riesz-Haviland Theorem Structure of Positive Polynomials Cubic Column Relations Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 2 / 57

General Idea to Study TMP TMP is more general than FMP: fewer moments = ⇒ less data Stochel: link between TMP and FMP Existing approaches are directed at enlarging the data by acquiring new moments, and eventually making the problem into one of flat data type (i.e., with intrinsic recursiveness). This naturally leads to a full MP. If such a flat extension of the initial data cannot be accomplished, then TMP has no representing measure. H elpful tool: Smul’jan’s Theorem on positivity of 2 × 2 matrices Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 3 / 57

The Classical (Full) Moment Problem Let β ≡ β ( ∞ ) = { β i } i ∈ Z d + denote a d -dimensional real multisequence, and let K (closed) ⊆ R d . The (full) K -moment problem asks for necessary and sufficient conditions on β to guarantee the existence of a positive Borel measure µ supported in K such that � x i d µ ( i ∈ Z d β i = + ); µ is called a rep. meas. for β. Associated with β is a moment matrix M ≡ M ( ∞ ), defined by ( i , j ∈ Z d M ij := β i + j + ) . Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 4 / 57

Basic Positivity Condition P n : polynomials p over R with deg p ≤ n 0 ≤ i + j ≤ n a i x i , Given p ∈ P n , p ( x ) ≡ � � p ( x ) 2 d µ ( x ) 0 ≤ � � x i + j d µ ( x ) = � = a i a j a i a j β i + j . ij ij Now recall that we’re working in d real variables. To understand this “matricial” positivity , we introduce the following lexicographic order on the rows and columns of M ( n ): 1 , X 1 , . . . , X d , X 2 1 , X 2 X 1 , . . . , X 2 d , . . . Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 5 / 57

Also recall that M ( n ) i , j := β i + j . Then � ( “matricial” positivity) a i a j β i + j ≥ 0 ij ⇔ M ( n ) ≡ M ( n )( β ) ≥ 0 . Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 6 / 57

For example, for moment problems in R 2 ,   β 00 β 01 β 10   M (1) =  , β 01 β 02 β 11    β 10 β 11 β 20   β 00 β 01 β 10 β 02 β 11 β 20   β 01 β 02 β 11 β 03 β 12 β 21      β 10 β 11 β 20 β 12 β 21 β 30    M (2) = .   β 02 β 03 β 12 β 04 β 13 β 22       β 11 β 12 β 21 β 13 β 22 β 31     β 20 β 21 β 30 β 22 β 31 β 40 Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 7 / 57

In general, � � M ( n ) B M ( n + 1) = B ∗ C Similarly, one can build M ( ∞ ) ≡ M ( ∞ )( β ) ≡ M ( β ). The link between TMP and FMP is provided by a result of Stochel (2001): Theorem (Stochel’s Theorem) β ( ∞ ) has a rep. meas. supported in a closed set K ⊆ R d if and only if, for each n, β (2 n ) has a rep. meas. supported in K. Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 8 / 57

Moment Problems and Nonnegative Polynomials (FULL MP Case) M := { β ≡ β ( ∞ ) : β admits a rep. meas. µ } B + := { β ≡ β ( ∞ ) : M ( ∞ )( β ) ≥ 0 } Clearly, M ⊆ B + (Berg, Christensen and Ressel) β ∈ B + , β bounded ⇒ β ∈ M (Berg and Maserick) β ∈ B + , β exponentially bounded ⇒ β ∈ M (RC and L. Fialkow) β ∈ B + , M ( β ) finite rank ⇒ β ∈ M Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 9 / 57

P + : nonnegative poly’s Σ 2 : sums of squares of poly’s Clearly, Σ 2 ⊆ P + Duality For C a cone in R Z d + , we let C ∗ := { ξ ∈ R Z d + : supp( ξ ) is finite and � p , ξ � ≥ 0 for all p ∈ C } . (Riesz-Haviland) P ∗ + = M For, consider the Riesz functional Λ β ( p ) := p ( β ) ≡ � p , β � , which induces a map M → P ∗ + ( β �→ Λ β ); Haviland’s Theorem says that this maps is onto, that is, ∃ µ rep. meas. for β ⇔ Λ β ≥ 0 on P + . Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 10 / 57

P + = M ∗ (straightforward once we have a r.m.) B + = (Σ 2 ) ∗ (straightforward) (Berg, Christensen and Jensen) ( B + ) ∗ = Σ 2 ( n = 1) P + = Σ 2 ⇒ P ∗ + = (Σ 2 ) ∗ ⇒ M = B + (Hamburger) Generally, SOS implies the existence of a representing measure. Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 11 / 57

Localizing Matrices Consider the full, complex MP � z i z j d µ = γ ij ( i , j ≥ 0) , ¯ where supp µ ⊆ K , for K a closed subset of C . The Riesz functional is given by z i z j ) := γ ij ( i , j ≥ 0) . Λ γ (¯ Riesz-Haviland: There exists µ with supp µ ⊆ K ⇔ Λ γ ( p ) ≥ 0 for all p such that p | K ≥ 0 . Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 12 / 57

If q is a polynomial in z and ¯ z , and K ≡ K q := { z ∈ C : q ( z , ¯ z ) ≥ 0 } , then L q ( p ) := L ( qp ) must satisfy L q ( p ¯ p ) ≥ 0 for µ to exist. For, � L q ( p ¯ p ) = qp ¯ p d µ ≥ 0 (all p ) . K q K. Schm¨ udgen (1991): If K q is compact, Λ γ ( p ¯ p ) ≥ 0 and L q ( p ¯ p ) ≥ 0 for all p , then there exists µ with supp µ ⊆ K q . Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 13 / 57

The Truncated Complex Moment Problem Given γ : γ 00 , γ 01 , γ 10 , . . . , γ 0 , 2 n , . . . , γ 2 n , 0 , with γ 00 > 0 and γ ji = ¯ γ ij , the TCMP entails finding a positive Borel measure µ supported in the complex plane C such that � z i z j d µ γ ij = ¯ (0 ≤ i + j ≤ 2 n ); 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 . Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 14 / 57

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. Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 15 / 57

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 = α 1 + β Z and Z ) (1 ≤ k ≤ [ n Z k = p k − 1 ( Z , ¯ 2] + 1; deg p k − 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 R d . Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 16 / 57

Applications Subnormal Operator Theory (unilateral weighted shifts) Physics (determination of contours) Computer Science (image recognition and reconstruction) Geography (location of proposed distribution centers) Probability (reconstruction of p.d.f.’s) Environmental Science (oil spills, via quadrature domains) Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 17 / 57

Engineering (tomography) Geophysics (inverse problems, cross sections) Typical Problem : Given a 3-D body, let X-rays act on the body at different angles, collecting the information on a screen. One then seeks to obtain a constructive, optimal way to approximate the body, or in some cases to reconstruct the body. Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 18 / 57

Positivity of Block Matrices Theorem (Smul’jan, 1959)  A ≥ 0  � �  A B  ≥ 0 ⇔ B = AW . B ∗ C  C ≥ W ∗ AW   � � A B =rank A ⇔ C = W ∗ AW . Moreover, rank B ∗ C Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 19 / 57

Corollary � � A B Assume rank = rank A. Then B ∗ C � � A B A ≥ 0 ⇔ ≥ 0 . B ∗ C Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 20 / 57

Basic Positivity Condition P n : polynomials p in z and z , deg p ≤ n z i z j , Given p ∈ P n , p ( z , z ) ≡ � 0 ≤ i + j ≤ n a ij ¯ � | p ( z , z ) | 2 d µ ( z , z ) 0 ≤ � � z i + ℓ z j + k d µ ( z , z ) = a ij ¯ a k ℓ ¯ ijk ℓ � = a ij ¯ a k ℓ γ i + ℓ, j + k . ijk ℓ 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 , . . . Ra´ ul Curto (Helton Wrkshp, 10/04/10) TMP with Finite Varieties 21 / 57