On truncated discrete moment problems Tobias Kuna University of - PowerPoint PPT Presentation

On truncated discrete moment problems Tobias Kuna University of Reading, UK (Joint work with Maria Infusino, Joel Lebowitz, Eugene Speer) IWOTA 2019 Lisbon – 26th July T. Kuna On truncated discrete moment problems

Discrete truncated moment problem This talk focus on K discrete subset of R d for d = 1 ; n ∈ N or d ≥ 2 , n = 2 mainly K = N 0 or K = Z d . d − dimensional truncated K − moment problem of degree n � m ( 0 ) , . . . , m ( n ) � with m ( k ) a tuple Given m : = � � m ( k ) j 1 , ... , j d j r ∈ N 0 ; ∑ d r = 1 j r = k with m ( k ) j 1 , ... , j d ∈ R . Find a nonnegative Radon measure µ supported in K s.t. � m ( k ) K x j 1 1 . . . x j d ∀ k ; j r ∈ N 0 with ∑ j 1 , ... , j d = d µ ( dx ) , j r = n r W.l.o.g. we can assume m 0 = 1 and µ is a probability measure on K . We can use that the set is discrete m ( k ) j 1 , ... , j d = ∑ x j 1 1 . . . x j d d µ ( { x } ) , x ∈ K T. Kuna On truncated discrete moment problems

Motivation for the discrete TMP Main motivation (for me) Moment problem for point processes Complex systems, Material science, Statistical mechanics Point processes Let R be a Riemannian manifold. � � δ r i ∈ D ′ ( R ) : I countable and r i ∈ R ∈ ⊂ D ′ ( R ) K : = ∑ i ∈ I A measure µ on K is called a point process. K is infinite dimensional d = ∞ . all element of K are Radon measures. Interpretation: µ is probability to find point configuration η . T. Kuna On truncated discrete moment problems

Relation to N d 0 -TMP For η = ∑ i ∈ I δ r i ∈ K , define N A ( η ) : = η ( A ) = number of points in η which are in A By definition N A : K → N 0 . Finite dimensional distribution of µ One-dimensional distributions µ A : µ A ( C ) : = µ ( { η : N A ( η ) ∈ C } ) Push-forward of µ w.r.t. N A . Two-dimensional distribution µ A 1 , A 2 given by µ A 1 , A 2 ( C 1 × C 2 ) : = µ ( { η : N A i ( η ) ∈ C i } ) and so on Support of µ A is N 0 . Support of µ A 1 , A 2 is N 0 × N 0 . and so on T. Kuna On truncated discrete moment problems

General convex analysis Generalized Tchakaloff Thm (Richter-Bayer-Teichmann) m has a N 0 − representing measure � criterion to solve { 0 , 1 , . . . , N } -TMP ∃ N ∈ N s.t. m has a { 0 , 1 , . . . , N }− representing measure criterion to solve N 0 -TMP depending on (unbeknown) N T. Kuna On truncated discrete moment problems

Solving { 0 , 1 , . . . , N } -TMP Fix n , N ∈ N s.t. N ≥ n . Aim: Characterize the set S N of all n − tuple admitting m = ( m 1 , . . . , m n ) ∈ R n a { 0 , 1 , . . . , N } -representing probability measures. Every { 0 , 1 , . . . , N } -representing probability for m is a convex combination of probabilities concentrated at k = 0 , 1 , . . . , N . Hence S N is the convex hull of A N : = { ( k , k 2 , . . . , k n ) | k = 0 , 1 , . . . , N } Classical convex analysis yields, that S N is the intersection of finitely many closed half-spaces H containing A N whose � ∂ H contains at least n points � bounding hyperplanes ∂ H ↔ from A N   polynomials of degree n with leading coefficient ± 1   bounding hyperplanes ∂ H ↔   n distinct roots in { 0 , 1 , . . . , N }   nonnegative on { 0 , 1 , . . . , N } T. Kuna On truncated discrete moment problems

Solving { 0 , 1 , . . . , N } -TMP   polynomials of degree n with leading coefficient + 1   P n , N : =   n distinct roots in { 0 , 1 , . . . , N }   nonnegative on { 0 , 1 , . . . , N } If n = 2 j even, any P ∈ P n , N is of the form: P ( x ) = � x − k 1 �� x − ( k 1 + 1 ) � � x − k j �� x − ( k j + 1 ) � . . . with zeros k 1 < k 1 + 1 < k 2 < k 2 + 1 < . . . < k j in { 0 , 1 , . . . , N } . If n = 2 j + 1 odd, any P ∈ P n , N is of the form: � x − k 1 �� x − ( k 1 + 1 ) � � x − k j �� x − ( k j + 1 ) � P ( x ) = x . . . with zeros 0 < k 1 < k 1 + 1 < k 2 < k 2 + 1 < . . . < k j in { 0 , 1 , . . . , N } .   polynomials of degree n with leading coefficient − 1   Q n , N : = = { P ( x )( N − x ) | P ∈ P n − 1 , N − 1 }   n distinct roots in { 0 , 1 , . . . , N }   nonnegative on { 0 , 1 , . . . , N } T. Kuna On truncated discrete moment problems

From { 0 , 1 , . . . , N } -TMP to N 0 -TMP Generalized Tchakaloff Thm (Richter-Bayer-Teichmann) criterion to solve { 0 , 1 , . . . , N } -TMP m has a N 0 − representing probability m has a { 0 , 1 , . . . , N } -repr. prob. � ∃ N ∈ N s.t. m has a � { 0 , 1 , . . . , N }− representing probability L m ( p ) ≥ 0 , ∀ p ∈ P n , N ∪ Q n , N first criterion to solve N 0 -TMP m has a N 0 − representing probability � ∃ N ∈ N s.t. L m ( p ) ≥ 0 for all p ∈ P n , N ∪ Q n , N Problem: How to identify or get rid of N ? T. Kuna On truncated discrete moment problems

N independent condition Note that P n , N ⊂ P n , N + 1 Define P n : = � N ∈ N P n , N . � ⇒ � � � m has a N 0 -representing measure L m ( p ) ≥ 0 ∀ p ∈ P n . Recall � m has a { 0 , 1 , . . . , N } -repr. prob. � ⇔ � � m has a N 0 -repr. prob. for some N large enough The condition L m (( M − x ) p ) ≥ 0 ML m ( p ) ≥ L m ( xp ) L m ( p ) ≥ 1 � � � L m ( p ) ≥ 0 ∀ p ∈ Q n , M ⇔ ML which implies that L m ( p ) ≥ 0 , ∀ p ∈ P n − 1 and if L m ( p ) = 0 for some p ∈ P n − 1 , then L m ( xp ) = 0 Necessary conditions L m ( p ) ≥ 0 , ∀ p ∈ P n ∪ P n − 1 m has a N 0 − repr.prob. ⇒ if L m ( p ) = 0 for some p ∈ P n − 1 then L m ( xp ) = 0 T. Kuna On truncated discrete moment problems

Theorem (Infusino, K., Lebowitz, Speer, 2017) L m ( p ) ≥ 0 , ∀ p ∈ P n ∪ P n − 1 m has a N 0 − repr.prob. ⇔ if L m ( p ) = 0 for some p ∈ P n − 1 then L m ( xp ) = 0 Moreover, non of the conditions can be dropped. Proof of ⇐ : One need to derive an a priori bound on N using only the above conditions not realizability. Previous results: Karlin and Studden 1966 on K = N 0 ∪ { ∞ } . Solvability condition depending on an unknown parameter The best one could hope to obtain using Semi-algebraic techniques is conditions p 2 � ≥ 0 �� x − k �� x − ( k + 1 ) � ∀ p polynomial and ∀ k ∈ N 0 L m Challenge: Can one reduce the conditions further by making them m dependent? T. Kuna On truncated discrete moment problems

m dependent conditions: what was known Case n = 1: � � � � m = ( m 1 ) is realizable ⇔ m 1 ≥ 0 Case n = 2: (Yamada 1961) � � m 2 − ( m 1 ) 2 ≥ ⌊ m 1 ⌋⌈ m 1 ⌉ � � m = ( m 1 , m 2 ) is realizable ⇒ Case n = 2: (K., Lebowitz, Speer 2009) m 1 > 0 ; m 2 − ( m 1 ) 2 ≥ ⌊ m 1 ⌋⌈ m 1 ⌉ � � � � m = ( m 1 , m 2 ) is realizable ⇔ or m 1 = 0 and m 2 = 0 Kwerel 1975, Prekopa et al. 1986: K = { 0 , 1 , . . . , N } some explicit (necessary) conditions for n = 2 , 3 but no explicit conditions for n ≥ 4. T. Kuna On truncated discrete moment problems

m dependent conditions We partition the set of all m : = ( m 1 , . . . , m n ) ∈ R n realizable on N 0 into: (i) m : = ( m 1 , . . . , m n ) is B-realisable if n � ∃ p ∈ P k with L m ( p ) = 0 k = 1 (ii) otherwise m is I -realisable , i.e. n � ∀ p ∈ P k one has L m ( p ) > 0 k = 1 Main Theorem (Infusino, K., Lebowitz, Speer, 2017) Let m : = ( m 1 , . . . , m n ) ∈ R n . If ( m 1 , . . . , m n − 1 ) is I-realisable, then ∃ p ( n ) ∈ P n s.t. p ( n ) does not m m depend on m n L m ( q ) ≥ L m ( p ( n ) ∀ q ∈ P n m ) , We call such a p ( n ) a minimizing polynomial for m . m Challenge: How to find p ( n ) m T. Kuna On truncated discrete moment problems

Finding p ( 2 ) m : case n = 2 Let m = ( m 1 , m 2 ) ∈ R 2 be such that m 1 is I-realisable, i.e. m 1 > 0. � � P 2 : = t k ( x ) : = ( x − k )( x − k − 1 ) | k ∈ N 0 Case n = 2, m 1 > 0 � � ⇔ m 2 − ( m 1 ) 2 ≥ ⌊ m 1 ⌋⌈ m 1 ⌉ ( m 1 , m 2 ) realisable on N 0 Case: n=2 P ( 2 ) � �� � m ( x ) = x − k x − ( k + 1 ) for k = ⌊ m 1 ⌋ corresponds to condition m 2 − ( m 1 ) 2 ≥ ⌊ m 1 ⌋⌈ m 1 ⌉ . Connection to Stieltjes TMP Case n = 2, m 1 > 0 � � � � 1 m 1 � � � ≥ 0 ⇔ m 2 − m 2 ( m 1 , m 2 ) realisable on [ 0 , + ∞ ) ⇔ 1 ≥ 0 � � m 1 m 2 � T. Kuna On truncated discrete moment problems

Connection between N 0 − TMP & [ 0 , + ∞ ) − TMP Let m = ( m 1 , . . . , m n − 1 , m n ) ∈ R n s.t. ( m 1 , . . . , m n − 1 ) is I-realizable on N 0 ⇓ ( m 1 , . . . , m n − 1 ) is I-realizable on [ 0 , + ∞ ) m n ∈ R s.t. ˆ m : = ( m 1 , . . . , m n − 1 , ˆ m n ) is realizable on [ 0 , + ∞ ) Take the smallest ˆ Curto-Fialkow 1991 ⇓ m is B-realizable on [ 0 , + ∞ ) ˆ m has a unique [ 0 , + ∞ ) − representing probability ν ˆ the support of ν is given by the zeros of a polynomial determined only by ( m 1 , . . . , m n − 1 ) . n = 2 : supp ( ν ) = { m 1 } n = 3 : supp ( ν ) = { 0 , m 2 / m 1 } zeros of p ( 2 ) n = 2 : supp ( ν ) = { m 1 } , m = {⌊ m 1 ⌋ , ⌊ m 1 ⌋ + 1 } ; zeros of p ( 2 ) supp ( ν ) = { 0 , m 2 / m 1 } , m = { 0 , ⌊ m 2 / m 1 ⌋ , ⌊ m 2 / m 1 ⌋ + 1 } . n = 3 : Conjecture The zeros of p ( n ) are the nearest integers to the points in supp ( ν ) m T. Kuna On truncated discrete moment problems True for n = 2 , 3 but false for n ≥ 4 !