Projective limit techniques for the infinite dimensional moment - - PowerPoint PPT Presentation
Projective limit techniques for the infinite dimensional moment problem Maria Infusino University of Konstanz (Joint work with S. Kuhlmann, T. Kuna, P. Michalski) EWM-GM 2018 Karl-Franzens-Universitt Graz September 7th, 2018 Maria
Maria Infusino
University of Konstanz (Joint work with S. Kuhlmann, T. Kuna, P. Michalski)
EWM-GM 2018 Karl-Franzens-Universität Graz – September 7th, 2018
Maria Infusino Projective limit techniques for infinite dim. MP
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach
1
Motivation and Framework The classical full K−Moment Problem (KMP) A general formulation of the full KMP
2
Our strategy for solving the general KMP Preliminaries on projective limits The character space as a projective limit
3
Outcome of our "projective limit" approach Old and new results for the KMP Final remarks and open questions
Maria Infusino Projective limit techniques for infinite dim. MP
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach The classical full K−Moment Problem (KMP) A general formulation of the full KMP
Let µ be a nonnegative Borel measure defined on R. The n−th moment of µ is: mµ
n :=
Z
R
xnµ(dx) If all moments of µ exist and are finite, then (mµ
n )∞ n=0 is the moment sequence of µ.
µ non-neg. Borel measure with all moments finite Moment Sequence of µ
?
Let N ∈ N ∪ {∞} and K ⊆ R closed. The one-dimensional K−Moment Problem (KMP) Given a sequence m = (mn)N
n=0 of real numbers, does there exist a nonnegative Radon
measure µ supported on a closed K ⊆ R s.t. for any n = 0, 1, . . . , N we have mn = Z
K
xnµ(dx) | {z }
n-th moment of µ
? N = ∞ Full KMP N ∈ N Truncated KMP
Maria Infusino Projective limit techniques for infinite dim. MP 1 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach The classical full K−Moment Problem (KMP) A general formulation of the full KMP
Riesz’s Functional Let m = (mn)∞
n=0 be such that mn 2 R.
Lm : R[x] ! R p(x) :=
N
P
n=0
an xn 7! Lm(p) :=
N
P
n=0
an mn. Note: If m is represented by a nonnegative measure µ on K, then Lm(p) =
N
X
n=0
an mn =
N
X
n=0
an Z
K
xnµ(dx) = Z
K
p(x)µ(dx). The one dimensional KMoment Problem (KMP) Given a sequence m = (mn)∞
n=0 of real numbers, does there exist a nonnegative Radon
measure µ supported on a closed K ✓ R s.t. for any p 2 R[x] we have Lm(p) = Z
K
p(x)µ(dx) ? The one-dimensional KMoment Problem (KMP)
Maria Infusino Projective limit techniques for infinite dim. MP 2 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach The classical full K−Moment Problem (KMP) A general formulation of the full KMP
Let x := (x1, . . . , xd) with d 2 N. The d-dimensional KMoment Problem (KMP) Given a linear functional L : R[x] ! R, does there exist a nonnegative Radon measure µ supported on a closed K ✓ Rd s.t. for any p 2 R[x] we have L(p) = Z
K
p(x)µ(dx) ? What if we have infinitely many real variables? What if we take a generic Rvector space V (even inf. dim.) instead of Rd? What if we take a generic unital commutative Ralgebra A instead of R[x] ? Infinite dimensional K-Moment Problem
Maria Infusino Projective limit techniques for infinite dim. MP 3 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach The classical full K−Moment Problem (KMP) A general formulation of the full KMP
Terminology and Notations: A = unital commutative R−algebra X(A)=character space of A=Hom(A; R) For a ∈ A the Gelfand transform ˆ a : X(A) → R is ˆ a(α) := α(a), ∀α ∈ X(A). X(A) is given the weakest topology τA s.t. all ˆ a, a ∈ A are continuous. The K−moment problem for unital commutative R−algebras Given a linear functional L : A → R, does there exist a nonnegative Radon measure µ supported on a closed subset K ⊆ X(A) s.t. for any a ∈ A we have L(a) = Z
X(A)
ˆ a(α)µ(dα) ? If yes, µ is called K−representing (Radon) measure for L. Recall that µ is supported on K ⊆ X(A) if µ(X(A) \ K) = 0. NB: Finite dimensional MP is a particular case If A = R[x] = R[x1, . . . , xd] then X(A) = X(R[x]) is identified (as tvs) with Rd.
Maria Infusino Projective limit techniques for infinite dim. MP 4 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach The classical full K−Moment Problem (KMP) A general formulation of the full KMP
Terminology and Notations: A = unital commutative R−algebra X(A)=character space of A=Hom(A; R) For a ∈ A the Gelfand transform ˆ a : X(A) → R is ˆ a(α) := α(a), ∀α ∈ X(A). X(A) is given the weakest topology τA s.t. all ˆ a, a ∈ A are continuous. The K−moment problem for R−algebras Given a linear functional L : R[x] → R, does there exist a nonnegative Radon measure µ supported on a closed K ⊆ X(R[x]) = Rd s.t. for any a ∈ R[x] we have L(a) = Z
X(R[x])
ˆ a(α)µ(dα) = Z
Rd a(α)µ(dα) ?
Recall that µ is supported on K ⊆ Rd if µ( Rd \K) = 0. NB: Finite dimensional KMP is a particular case If A = R[x] = R[x1, . . . , xd] then X(A) = X(R[x]) is identified (as tvs) with Rd.
Maria Infusino Projective limit techniques for infinite dim. MP 4 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Preliminaries on projective limits The character space as a projective limit
The K−moment problem for unital commutative R−algebras Given a linear functional L : A → R, does there exist a nonnegative Radon measure µ supported on a closed subset K ⊆ X(A) s.t. for any a ∈ A we have L(a) = Z
X(A)
ˆ a(α)µ(dα) ? If yes, µ is called K−representing (Radon) measure for L. Our idea construct X(A) as a projective limit of all (X(S), BS)
finite dimensional moment theory existence criteria for X(A)-representing cylindrical measures extension theorems for cylindrical measures existence criteria for X(A)-representing Radon measures
Maria Infusino Projective limit techniques for infinite dim. MP 5 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Preliminaries on projective limits The character space as a projective limit
(I, ) directed partially ordered set {(Xi, Σi), ⇡i,j, I} projective system of measurable spaces, i.e. (Xi, Σi) measurable spaces ⇡i,j : Xj ! Xi defined and measurable 8 i j in I s.t. Xk Xj Xi
πj,k πi,k πi,j
Projective limit of {(Xi, Σi), ⇡i,j, I} is a measurable space (XI , ΣI ) together with maps ⇡i : XI ! Xi for i 2 I s.t. ⇡i,j ⇡j = ⇡i for all i j in I ΣI is the smallest algebra w.r.t. which all ⇡i’s are measurable For any measurable space (Y , ΣY ) and any measurable fi : Y ! Xi with i 2 I and fi = ⇡i,j fj, 8 i j, 9! measurable f : Y ! XI s.t. ⇡i f = fi 8 i 2 I. Y XI Xj Xi
f fi fj πj πi πi,j Maria Infusino Projective limit techniques for infinite dim. MP 6 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Preliminaries on projective limits The character space as a projective limit
P := {(Xi, Σi), ⇡i,j, I} = projective system of measurable spaces {(XI , ΣI ), ⇡i, I} = projective limit of P XI Xj Xi
πj πi πi,j
cylinder set in XI : ⇡−1
i
(M) for some i 2 I and M 2 Σi cylinder algebra on XI : CI := {⇡−1
i
(M) : M 2 Σi, 8 i 2 I} . cylinder algebra on XI : (CI ) ⌘ ΣI Cylindrical quasi-measure A cylindrical quasi-measure µ w.r.t. P is a set function µ : CI ! R+ s.t. ⇡i #µ is a measure on Σi for all i 2 I. NB: Cylindrical quasi-measures are NOT measures! Question 1 When can a cylindrical quasi-measure w.r.t. P be extended to a measure on (XI , ΣI )?
Maria Infusino Projective limit techniques for infinite dim. MP 7 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Preliminaries on projective limits The character space as a projective limit
(I, ) directed partially ordered set {(Xi, ⌧i), ⇡i,j, I} projective system of topological spaces, i.e. (Xi, ⌧i) topological spaces ⇡i,j : Xj ! Xi defined and continuous 8 i j in I s.t. Xk Xj Xi
πj,k πi,k πi,j
Projective limit of {(Xi, ⌧i), ⇡i,j, I} is a topological space (XI , ⌧I ) together with maps ⇡i : XI ! Xi for i 2 I s.t. ⇡i,j ⇡j = ⇡i for all i j in I ⌧I is the weakest topology w.r.t. which all ⇡i’s are continuous For any topological space (Y , ⌧Y ) and any continuous fi : Y ! Xi with i 2 I and fi = ⇡i,j fj, 8 i j, 9! continuous f : Y ! XI s.t. ⇡i f = fi 8 i 2 I. Y XI Xj Xi
f fi fj πj πi πi,j Maria Infusino Projective limit techniques for infinite dim. MP 8 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Preliminaries on projective limits The character space as a projective limit
(I, ) directed partially ordered set T := {(Xi, ⌧i), ⇡i,j, I} projective system of Hausdorff topological spaces PT := {(Xi, Bi), ⇡i,j, I} associated projective system of Borel measurable spaces Bi := Borel algebra on Xi w.r.t. ⌧i (XI , ΣI ) (Xj, Bj) (Xi, Bi)
πj πi πi,j
(XI , ⌧I ) (Xj, ⌧j) (Xi, ⌧i)
πj πi πi,j
(XI , CI ) , ! (XI , ΣI ) , ! (XI , BI ) Cylindrical Cylindrical measure Borel measure quasi-measure Question 1 When can a cylindrical quasi-measure be extended to a measure on (XI , ΣI )? Question 2 When can a cylindrical quasi-measure be extended to a Radon measure on (XI , BI )?
Maria Infusino Projective limit techniques for infinite dim. MP 9 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Preliminaries on projective limits The character space as a projective limit
(I, ) directed partially ordered set T := {(Xi, ⌧i), ⇡i,j, I} projective system of Hausdorff topological spaces PT := {(Xi, Bi), ⇡i,j, I} associated projective system of Borel measurable spaces An exact projective system of Radon measures w.r.t. PT is a family {µi, i 2 I} s.t.
cylindrical quasi-measure , exact projective system of measures µ(⇡−1
i
(Ei)) = µi(Ei), 8i 2 I, 8Ei 2 Bi Answer to Question 1 (Prokhorov, 1956) If {µi, i 2 I} is an exact projective system of Radon probabilities w.r.t. PT , then 9! cylinder probability ⌫ on (XI , ΣI ) such that ⇡i #⌫ = µi for all i 2 I. Answer to Question 2 (Prokhorov, 1956) If {µi, i 2 I} is an exact projective system of Radon probabilities w.r.t. PT , then 9! Radon probability µ on (XI , BI ) such that ⇡i #µ = µi for all i 2 I if and only if 8" > 0 9 K ⇢ XI compact s.t. 8i 2 I, µi(⇡i(K)) 1 " (UT)
Maria Infusino Projective limit techniques for infinite dim. MP 10 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Preliminaries on projective limits The character space as a projective limit
A = unital commutative Ralgebra X(A)=character space of A=Hom(A; R) For a 2 A the Gelfand transform ˆ aA : X(A) ! R is ˆ aA(↵) := ↵(a), 8↵ 2 X(A). X(A) is given the weakest topology ⌧A s.t. all ˆ a, a 2 A are continuous. For any S, T subalgebras of A s.t. S ✓ T, we define ⇡S,T : X(T) ! X(S) ↵ 7! ↵ S then 8S ✓ T ✓ R subalgebras of A: X(R) X(T) X(S)
πT,R πS,R πS,T
J := {S ✓ A : S finitely generated subalgebra of A, 1 2 S} directed partially ordered set If for any S 2 J: ⌧S :=the weakest topology ⌧S on X(S) s.t. all ˆ aS, a 2 S are continuous. BS be the Borel –algebra on X(S) w.r.t. ⌧S then {(X(S), ⌧S), ⇡S,T , J} is a projective system of Hausdorff topological spaces {(X(S), BS), ⇡S,T , J} is a projective system of Borel measurable spaces
Maria Infusino Projective limit techniques for infinite dim. MP 11 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Preliminaries on projective limits The character space as a projective limit
A = unital commutative Ralgebra J := {S ✓ A : S finitely generated subalgebra of A, 1 2 S}
Proposition {(X(A), τA), πS, J} is the projective limit of {(X(S), τS), πS,T, J} {(X(A), ΣJ), πS, J} is the projective limit of {(X(S), BS), πS,T, J} where for any S 2 J πS := πS,A : X(A) ! X(S), α 7! α S ΣJ the smallest σ–algebra on X(A) s.t. all the πS, S 2 J are measurable
(X(A), ΣJ ) (X(T), BT ) (X(S), BS )
πT πS πS,T
(X(A), τJ = τA) (X(T), τT ) (X(S), τS )
πT πS πS,T
(X(A), ΣJ) , ! (X(A), BJ) Representing cylindrical measure Representing Radon measure
Maria Infusino Projective limit techniques for infinite dim. MP 12 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Old and new results for the KMP Final remarks and open questions
Theorem* (I., Kuhlmann, Kuna, Michalski, 2018) Let A be a unital commutative Ralgebra, L : A ! R s.t. L(1) = 1 and J := {S ✓ A : S finitely generated subalgebra of A, 1 2 S}. ✓ 8S 2 J, 9! X(S)representing Radon measure µS for L S ◆ = ) ✓ 9! X(A)representing cylindrical measure µ for L ◆ Sketch of the proof P := {(X(S), BS), ⇡S,T , J} projective system ⌫ is a constructibly Radon measure [Ghasemi-Kulmann-Marshall, ’16] + 8S ✓ T in J both µS and ⇡S,T #µT are X(S)–representing Radon measures for L S + UNIQUENESS HP 8S ✓ T, µS ⌘ ⇡S,T #µT 8 S 2 J, µS(XS) = L S (1) = L(1) = 1
+ THM 1 (Prokhorov) 9!⌫ measure on (X(A), ΣJ) s.t. ⇡S #⌫ = µS, 8 S 2 J Hence, for any a 2 A we have a 2 S for some S 2 J and so L(a) = L S (a) = Z
X(S)
ˆ a()dµS() = Z
X(A)
ˆ a(⇡S())d⌫() = Z
X(A)
ˆ a(↵)d⌫(↵). + THM 1 (Prokhorov)
Maria Infusino Projective limit techniques for infinite dim. MP 13 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Old and new results for the KMP Final remarks and open questions
Theorem (I., Kuhlmann, Kuna, Michalski, 2018) Let A be a unital commutative R−algebra, L : A → R s.t. L(1) = 1 and if (I.) L(a2) ≥ 0 for all a ∈ A. (II.) For each a ∈ A, the class C{ p L(a2n)} is quasi-analytic. then ∃! X(A)−representing cylindrical measure ν on (X(A), ΣJ) for L. Theorem (Nussbaum, 1965) Let L : R[X1, . . . , Xd] → R be linear s.t. L(1) = 1. If (i) L(p2) ≥ 0 for all p ∈ R[X1, . . . , Xd]. (ii) ∀i = 1, . . . , d : P∞
n=1 1
2n
q L(X 2n
i
) = ∞ Carleman Condition
then ∃! Rd−representing Radon measure for L.
Maria Infusino Projective limit techniques for infinite dim. MP 14 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Old and new results for the KMP Final remarks and open questions
Theorem (I., Kuhlmann, Kuna, Michalski, 2018) Let A be a unital commutative R−algebra, L : A → R s.t. L(1) = 1 and if (I.) L(a2) ≥ 0 for all a ∈ A. (II.) For each a ∈ A, the class C{ p L(a2n)} is quasi-analytic. then ∃! X(A)−representing cylindrical measure ν on (X(A), ΣJ) for L. Special case: A = R[Xi, i ∈ Ω] with Ω arbitrary index set. Theorem (Ghasemi, Kuhlmann, Marshall, 2016), similar result in (Schmüdgen, 2018) Let L : R[Xi, i ∈ Ω] → R be linear s.t. L(1) = 1. If (i) L(p2) ≥ 0 for all p ∈ R[Xi, i ∈ Ω]. (ii) ∀i ∈ Ω : P∞
n=1 1
2n
q L(X 2n
i
) = ∞.
then ∃! RΩ−representing constructibly Radon measure for L.
Maria Infusino Projective limit techniques for infinite dim. MP 14 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Old and new results for the KMP Final remarks and open questions
Theorem (I., Kuhlmann, Kuna, Michalski, 2018) Let A be a unital commutative R−algebra, L : A → R s.t. L(1) = 1 and if (I.) L(a2) ≥ 0 for all a ∈ A. (II.) For each a ∈ A, the class C{ p L(a2n)} is quasi-analytic. then ∃! X(A)−representing cylindrical measure ν on (X(A), ΣJ) for L. Special case: A = R[Xi, i ∈ Ω] with Ω arbitrary index set. Theorem (Ghasemi, Kuhlmann, Marshall, 2016), similar result in (Schmüdgen, 2018) Let L : R[Xi, i ∈ Ω] → R be linear s.t. L(1) = 1. If Ω is countable and (i) L(p2) ≥ 0 for all p ∈ R[Xi, i ∈ Ω]. (ii) ∀i ∈ Ω : P∞
n=1 1
2n
q L(X 2n
i
) = ∞.
then ∃! RΩ−representing ((((
( hhhh h
constructibly Radon measure for L.
Maria Infusino Projective limit techniques for infinite dim. MP 14 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Old and new results for the KMP Final remarks and open questions
Theorem** (I., Kuhlmann, Kuna, Michalski, 2018) Let A be a unital commutative Ralgebra, L : A ! R s.t. L(1) = 1 and J := {S ✓ A : S finitely generated subalgebra of A, 1 2 S}. ✓ 8S 2 J, 9! X(S)representing Radon measure µS for L S + (UT) ◆ = ) ✓ 9! X(A)representing Radon measure µ for L ◆ Sketch of the proof P := {(X(S), BS), ⇡S,T , J} projective system + 8S ✓ T in J both µS and ⇡S,T #µT are X(S)–representing Radon measures for L S + UNIQUENESS HP 8S ✓ T, µS ⌘ ⇡S,T #µT 8 S 2 J, µS(XS) = L S (1) = L(1) = 1
+ THM 2 (Prokhorov) 9!⌫ Radon measure on (X(A), BJ) s.t. ⇡S #⌫ = µS, 8 S 2 J Hence, for any a 2 A we have a 2 S for some S 2 J and so L(a) = L S (a) = Z
X(S)
ˆ a()dµS() = Z
X(A)
ˆ a(⇡S())d⌫() = Z
X(A)
ˆ a(↵)d⌫(↵).
Maria Infusino Projective limit techniques for infinite dim. MP 15 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Old and new results for the KMP Final remarks and open questions
Theorem Let A be a unital commutative Ralgebra, L : A ! R s.t. L(1) = 1. @ L(Q) ✓ [0, +1) for some Archimedean quadratic module Q of A,
i.e. ∀ a ∈ A, ∃ N ∈ N: N ± a ∈ M
1 A= ) ✓ 9! KQrepresenting Radon measure for L ◆ where KQ := {↵ 2 X(A) : ˆ q(↵) 0, 8q 2 Q}. This provides an alternative proof for the Jacobi-Prestel Positivstellensatz (2001). Theorem (Putinar, 1993) Let L : R[X1, . . . , Xd] ! R be linear s.t. L(1) = 1. ✓ L(Q) ✓ [0, +1) for some Archimedean quadratic module Q of A ◆ = ) ✓ 9! KQrepresenting Radon measure for L ◆ where KQ := {y 2 Rd : q(y) 0, 8q 2 Q} i.e. basic closed semi-algebraic set.
Maria Infusino Projective limit techniques for infinite dim. MP 16 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Old and new results for the KMP Final remarks and open questions
Theorem (I., Kuhlmann, Kuna, Michalski, 2018) Let A be a unital commutative Ralgebra, Q a quadratic module in A and L : A ! R s.t. L(1) = 1. If 9 Ba, Bc subalgebras of A such that Ba [ Bc generates A as a real algebra with Bc countably generated and (i) Q \ Ba is Archimedean in Ba (ii) For each a 2 Bc the class C{ p L(a2n)} is quasi-analytic (iii) L(Q) ✓ [0, +1) then 9! KQ-representing Radon measure with KQ := {↵ 2 X(A) :↵(q) 0, 8q 2 Q}. Theorem (Ghasemi, Kuhlmann, Marshall, 2016) Let Q be a quadratic module in R[Xi, i 2 Ω] and L : R[Xi, i 2 Ω] ! R be linear s.t. L(1) = 1. and . If 9 Λ ✓ Ω countable such that (i) Q \ R[Xi] is Archimedian for all i 2 Ω \ Λ. (ii) For each i 2 Λ, P∞
n=1 1
2n
q L(X 2n
i
) = 1
(iii) L(Q) ✓ [0, +1) then 9! KQ-representing Radon measure with KQ := {y 2 RΩ : q(y) 0, 8q 2 Q}.
Maria Infusino Projective limit techniques for infinite dim. MP 17 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Old and new results for the KMP Final remarks and open questions
Open questions Condition (ii) implies (UT). Does the converse hold? Does this approach allows to retrieve the known results about the KMP on the symmetric algebra of a locally convex space? Can our results be applied to localizations of a unital commutative real algebra (c.f. Marshall 2014, 2017) Advantages & Potential of the projective limit approach it is powerful technique to exploit the finite dimensional moment theory to get new advances in the infinite dimensional one. it provides a direct bridge from the KMP to a rich spectrum of tools coming from the theory of projective limits. it offers a unified setting in which compare the results known so far about the infinite dimensional KMP.
Maria Infusino Projective limit techniques for infinite dim. MP 18 / 18
Motivation and Framework Our strategy for solving the general KMP Outcome of our "projective limit" approach Old and new results for the KMP Final remarks and open questions
For more details see:
techniques for the infinite dimensional moment problem, soon on ArXiv!!
Maria Infusino Projective limit techniques for infinite dim. MP 18 / 18