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-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 Outline Motivation and Framework 1 The classical full K − Moment Problem (KMP) A general formulation of the full KMP Our strategy for solving the general KMP 2 Preliminaries on projective limits The character space as a projective limit Outcome of our "projective limit" approach 3 Old and new results for the KMP Final remarks and open questions Maria Infusino Projective limit techniques for infinite dim. MP
Motivation and Framework The classical full K − Moment Problem (KMP) Our strategy for solving the general KMP A general formulation of the full KMP Outcome of our "projective limit" approach The classical moment problem in one dimension Let µ be a nonnegative Borel measure defined on R . The n − th moment of µ is: Z x n µ ( dx ) m µ n := R If all moments of µ exist and are finite, then ( m µ n ) ∞ n = 0 is the moment sequence of µ . µ non-neg. Borel measure Moment Sequence of µ ? with all moments finite Let N ∈ N ∪ { ∞ } and K ⊆ R closed. The one-dimensional K − Moment Problem (KMP) Given a sequence m = ( m n ) 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 Z x n µ ( dx ) m n = ? K | {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 The classical full K − Moment Problem (KMP) Our strategy for solving the general KMP A general formulation of the full KMP Outcome of our "projective limit" approach Riesz’s Functional Riesz’s Functional Let m = ( m n ) ∞ n = 0 be such that m n 2 R . L m : R [ x ] ! R N N a n x n p ( x ) := P L m ( p ) := P 7! a n m n . n = 0 n = 0 Note: If m is represented by a nonnegative measure µ on K , then N N Z Z X X x n µ ( dx ) = L m ( p ) = a n m n = p ( x ) µ ( dx ) . a n K K n = 0 n = 0 The one dimensional K � Moment Problem (KMP) Given a sequence m = ( m n ) ∞ 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 Z L m ( p ) = p ( x ) µ ( dx ) ? K Maria Infusino Projective limit techniques for infinite dim. MP 2 / 18 The one-dimensional K � Moment Problem (KMP)
Motivation and Framework The classical full K − Moment Problem (KMP) Our strategy for solving the general KMP A general formulation of the full KMP Outcome of our "projective limit" approach The classical full finite dimensional K − moment problem Let x := ( x 1 , . . . , x d ) with d 2 N . The d -dimensional K � Moment Problem (KMP) Given a linear functional L : R [ x ] ! R , does there exist a nonnegative Radon measure µ supported on a closed K ✓ R d s.t. for any p 2 R [ x ] we have Z L ( p ) = p ( x ) µ ( d x ) ? K What if we have infinitely many real variables? What if we take a generic R � vector space V (even inf. dim.) instead of R d ? What if we take a generic unital commutative R � algebra A instead of R [ x ] ? Infinite dimensional K -Moment Problem Maria Infusino Projective limit techniques for infinite dim. MP 3 / 18
Motivation and Framework The classical full K − Moment Problem (KMP) Our strategy for solving the general KMP A general formulation of the full KMP Outcome of our "projective limit" approach 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 τ A 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 Z L ( a ) = a ( α ) µ ( d α ) ? ˆ X ( A ) 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 [ x 1 , . . . , x d ] then X ( A ) = X ( R [ x ]) is identified (as tvs) with R d . Maria Infusino Projective limit techniques for infinite dim. MP 4 / 18
Motivation and Framework The classical full K − Moment Problem (KMP) Our strategy for solving the general KMP A general formulation of the full KMP Outcome of our "projective limit" approach 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 τ A 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 ]) = R d s.t. for any a ∈ R [ x ] we have Z Z R d a ( α ) µ ( d α ) ? L ( a ) = a ( α ) µ ( d α ) = ˆ X ( R [ x ]) if µ ( R d \ K ) = 0. Recall that µ is supported on K ⊆ R d NB: Finite dimensional KMP is a particular case If A = R [ x ] = R [ x 1 , . . . , x d ] then X ( A ) = X ( R [ x ]) is identified (as tvs) with R d . Maria Infusino Projective limit techniques for infinite dim. MP 4 / 18
Motivation and Framework Preliminaries on projective limits Our strategy for solving the general KMP The character space as a projective limit Outcome of our "projective limit" approach Our strategy for solving the general KMP 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 Z L ( a ) = a ( α ) µ ( d α ) ? ˆ X ( A ) If yes, µ is called K − representing (Radon) measure for L . Our idea finite dimensional construct X ( A ) as a projective limit of all ( X ( S ) , B S ) moment theory • S finitely generated subalgebra of A with 1 ∈ S • B S Borel σ -algebra on X ( S ) w.r.t. τ S . extension theorems for existence criteria for X ( A ) -representing cylindrical measures cylindrical measures existence criteria for X ( A ) -representing Radon measures Maria Infusino Projective limit techniques for infinite dim. MP 5 / 18
Motivation and Framework Preliminaries on projective limits Our strategy for solving the general KMP The character space as a projective limit Outcome of our "projective limit" approach Projective limit of measurable spaces ( I , ) directed partially ordered set { ( X i , Σ i ) , ⇡ i , j , I } projective system of measurable spaces , i.e. X k ( X i , Σ i ) measurable spaces π j , k π i , k π i , j ⇡ i , j : X j ! X i defined and measurable 8 i j in I s.t. X j X i Projective limit of { ( X i , Σ i ) , ⇡ i , j , I } is a measurable space ( X I , Σ I ) together with maps ⇡ i : X I ! X i for i 2 I s.t. ⇡ i , j � ⇡ j = ⇡ i for all i j in I Σ I is the smallest � � algebra w.r.t. which all ⇡ i ’s are measurable For any measurable space ( Y , Σ Y ) and any measurable f i : Y ! X i with i 2 I and f i = ⇡ i , j � f j , 8 i j , 9 ! measurable f : Y ! X I s.t. ⇡ i � f = f i 8 i 2 I . Y f f j X I f i π j π i π i , j X j X i Maria Infusino Projective limit techniques for infinite dim. MP 6 / 18
Motivation and Framework Preliminaries on projective limits Our strategy for solving the general KMP The character space as a projective limit Outcome of our "projective limit" approach Cylindrical quasi-measures P := { ( X i , Σ i ) , ⇡ i , j , I } = projective system of measurable spaces { ( X I , Σ I ) , ⇡ i , I } = projective limit of P X I π j π i π i , j X j X i ⇡ − 1 cylinder set in X I : ( M ) for some i 2 I and M 2 Σ i i cylinder algebra on X I : C I := { ⇡ − 1 ( M ) : M 2 Σ i , 8 i 2 I } . i cylinder � � algebra on X I : � ( C I ) ⌘ Σ I Cylindrical quasi-measure A cylindrical quasi-measure µ w.r.t. P is a set function µ : C I ! R + s.t. ⇡ i # µ is a measure on Σ i 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 ( X I , Σ I ) ? Maria Infusino Projective limit techniques for infinite dim. MP 7 / 18
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