On the Stieltjes moment problem

Transcription

1 On the Stieltjes moment problem Jasson Vindas Department of Mathematics Ghent University Logic and Analysis Seminar March 11, 215

2 The problem of moments, as its generalizations, is an important mathematical problem which has attracted much attention for more than a century. It was first raised and solved by Stieltjes for positive measures. Problem (Stieltjes, 1894) Find conditions over {a n } n= which ensure the existence of solutions µ to the infinity system of equations where µ is a positive measure. x n dµ(x), n =, 1, 2,..., We will discuss several generalizations of this problem.

3 The problem of moments, as its generalizations, is an important mathematical problem which has attracted much attention for more than a century. It was first raised and solved by Stieltjes for positive measures. Problem (Stieltjes, 1894) Find conditions over {a n } n= which ensure the existence of solutions µ to the infinity system of equations where µ is a positive measure. x n dµ(x), n =, 1, 2,..., We will discuss several generalizations of this problem.

4 The classical Stieltjes moment problem Stieltjes found a necessary and sufficient condition for the existence of solutions. Define the sequence of matrices a a 1... a n a 1 a 2... a n+1 a 1 a 2... a n+1 a 2 a 3... a n+2 n =..... and (1) n = a n a n+1... a 2n a n+1 a n+2... a 2n+1 Theorem (Stieltjes, ) The Stieltjes moment problem has solution if and only if x n dµ(x), n =, 1, 2,..., det( n ) > and det( (1) n ) >, n =, 1, 2,....

5 The classical Stieltjes moment problem Stieltjes found a necessary and sufficient condition for the existence of solutions. Define the sequence of matrices a a 1... a n a 1 a 2... a n+1 a 1 a 2... a n+1 a 2 a 3... a n+2 n =..... and (1) n = a n a n+1... a 2n a n+1 a n+2... a 2n+1 Theorem (Stieltjes, ) The Stieltjes moment problem has solution if and only if x n dµ(x), n =, 1, 2,..., det( n ) > and det( (1) n ) >, n =, 1, 2,....

6 The classical Stieltjes moment problem Stieltjes found a necessary and sufficient condition for the existence of solutions. Define the sequence of matrices a a 1... a n a 1 a 2... a n+1 a 1 a 2... a n+1 a 2 a 3... a n+2 n =..... and (1) n = a n a n+1... a 2n a n+1 a n+2... a 2n+1 Theorem (Stieltjes, ) The Stieltjes moment problem has solution if and only if x n dµ(x), n =, 1, 2,..., det( n ) > and det( (1) n ) >, n =, 1, 2,....

7 Ideas connected with the Stieltjes moment problem Stieltjes influential papers led to many important ideas: The theory of Stieltjes integrals x n df (x), F. The Stieltjes transform, Re z / (, ], S(z) = df(x) x + z n= ( 1) n a n z n+1. Continued fraction approximations.

8 Ideas connected with the Stieltjes moment problem Stieltjes influential papers led to many important ideas: The theory of Stieltjes integrals x n df (x), F. The Stieltjes transform, Re z / (, ], S(z) = df(x) x + z n= ( 1) n a n z n+1. Continued fraction approximations.

9 Ideas connected with the Stieltjes moment problem Stieltjes influential papers led to many important ideas: The theory of Stieltjes integrals x n df (x), F. The Stieltjes transform, Re z / (, ], S(z) = df(x) x + z n= ( 1) n a n z n+1. Continued fraction approximations.

10 Ideas connected with the Stieltjes moment problem Stieltjes influential papers led to many important ideas: The theory of Stieltjes integrals x n df (x), F. The Stieltjes transform, Re z / (, ], S(z) = df(x) x + z n= ( 1) n a n z n+1. Continued fraction approximations.

11 Ideas connected with the Stieltjes moment problem Stieltjes influential papers led to many important ideas: The theory of Stieltjes integrals x n df (x), F. The Stieltjes transform, Re z / (, ], S(z) = df(x) x + z n= ( 1) n a n z n+1. Continued fraction approximations.

12 Modern approach goes back to Marcel Riesz (1921). Carleman ( ): connections with the theory of quasi-analytic functions. Other moment problems: Hamburger (192): Hausdorff (1923): c b x n df(x), n =, 1, 2,.... x n df (x), n =, 1, 2,.... For results on classical moment problems see the book by Shohat and Tamarkin (The problem of moments, 1943).

13 Modern approach goes back to Marcel Riesz (1921). Carleman ( ): connections with the theory of quasi-analytic functions. Other moment problems: Hamburger (192): Hausdorff (1923): c b x n df(x), n =, 1, 2,.... x n df (x), n =, 1, 2,.... For results on classical moment problems see the book by Shohat and Tamarkin (The problem of moments, 1943).

14 Modern approach goes back to Marcel Riesz (1921). Carleman ( ): connections with the theory of quasi-analytic functions. Other moment problems: Hamburger (192): Hausdorff (1923): c b x n df(x), n =, 1, 2,.... x n df (x), n =, 1, 2,.... For results on classical moment problems see the book by Shohat and Tamarkin (The problem of moments, 1943).

15 Moment problems for arbitrary sequences Theorem (Boas and Pólya, independently, 1939) Given an arbitrary sequence {a n } n=, there is always a function of bounded variation F such that x n df(x), n =, 1, 2,.... A. Durán achieved a major improvement to this result: Theorem (A. Durán, 1989) Every Stieltjes moment problem x n φ(x)dx, n =, 1, 2,..., admits a solution φ S(, ), namely, φ S(R) with supp φ [, ).

16 Moment problems for arbitrary sequences Theorem (Boas and Pólya, independently, 1939) Given an arbitrary sequence {a n } n=, there is always a function of bounded variation F such that x n df(x), n =, 1, 2,.... A. Durán achieved a major improvement to this result: Theorem (A. Durán, 1989) Every Stieltjes moment problem x n φ(x)dx, n =, 1, 2,..., admits a solution φ S(, ), namely, φ S(R) with supp φ [, ).

17 Moment problems for arbitrary sequences Theorem (Boas and Pólya, independently, 1939) Given an arbitrary sequence {a n } n=, there is always a function of bounded variation F such that x n df(x), n =, 1, 2,.... A. Durán achieved a major improvement to this result: Theorem (A. Durán, 1989) Every Stieltjes moment problem x n φ(x)dx, n =, 1, 2,..., admits a solution φ S(, ), namely, φ S(R) with supp φ [, ).

18 Stieltjes moment problems for arbitrary sequences A. Durán s proof: Laguerre expansions, Hankel transform. A. L. Durán and Estrada found a simple proof (1994): x n φ(x)dx, n =, 1, 2,..., (1) iff φ (n) () = ( i) n a n. Then, the Borel-Ritt theorem... Chung-Chung-Kim (23) exploited the method to show that (1) has solutions φ S β (, ), β > 1. Lastra and Sanz (29) have considered ultradifferentiable classes S (, ), with = (M p ), {M p }.

19 Stieltjes moment problems for arbitrary sequences A. Durán s proof: Laguerre expansions, Hankel transform. A. L. Durán and Estrada found a simple proof (1994): x n φ(x)dx, n =, 1, 2,..., (1) iff φ (n) () = ( i) n a n. Then, the Borel-Ritt theorem... Chung-Chung-Kim (23) exploited the method to show that (1) has solutions φ S β (, ), β > 1. Lastra and Sanz (29) have considered ultradifferentiable classes S (, ), with = (M p ), {M p }.

20 Stieltjes moment problems for arbitrary sequences A. Durán s proof: Laguerre expansions, Hankel transform. A. L. Durán and Estrada found a simple proof (1994): x n φ(x)dx, n =, 1, 2,..., (1) iff φ (n) () = ( i) n a n. Then, the Borel-Ritt theorem... Chung-Chung-Kim (23) exploited the method to show that (1) has solutions φ S β (, ), β > 1. Lastra and Sanz (29) have considered ultradifferentiable classes S (, ), with = (M p ), {M p }.

21 Stieltjes moment problems for arbitrary sequences A. Durán s proof: Laguerre expansions, Hankel transform. A. L. Durán and Estrada found a simple proof (1994): x n φ(x)dx, n =, 1, 2,..., (1) iff φ (n) () = ( i) n a n. Then, the Borel-Ritt theorem... Chung-Chung-Kim (23) exploited the method to show that (1) has solutions φ S β (, ), β > 1. Lastra and Sanz (29) have considered ultradifferentiable classes S (, ), with = (M p ), {M p }.

22 Abstract moment problem We want to replace x n φ(x)dx, n =, 1, 2,..., by the infinite system of linear equations f n, φ, n =, 1, 2,..., (2) where the sought solution φ is an element of a (topological!) vector space E and f n E. Problem Conditions over E and {f n } n= such that every generalized moment problem (4) has a solution φ E.

23 Abstract moment problem We want to replace x n φ(x)dx, n =, 1, 2,..., by the infinite system of linear equations f n, φ, n =, 1, 2,..., (2) where the sought solution φ is an element of a (topological!) vector space E and f n E. Problem Conditions over E and {f n } n= such that every generalized moment problem (4) has a solution φ E.

24 Abstract moment problem We want to replace x n φ(x)dx, n =, 1, 2,..., by the infinite system of linear equations f n, φ, n =, 1, 2,..., (2) where the sought solution φ is an element of a (topological!) vector space E and f n E. Problem Conditions over E and {f n } n= such that every generalized moment problem (4) has a solution φ E.

25 Particular cases The Borel problem: φ (n) (), n =, 1, 2,.... Here E = C (R) and f n = ( 1) n δ (n), elements of E (R). The Borel-Ritt problem. Given a sector S : α < arg z < β, z < r. Find an analytic function φ on S such that on any subsector S 1 : α 1 < arg z < β 1 one has φ(z) a n z n, z +. (3) n= In our setting, one may consider E the space of analytic functions on S having expansions of the form (3). The f n are the linear functionals sending ϕ to its n-th coefficient of the expansion.

26 Particular cases The Borel problem: φ (n) (), n =, 1, 2,.... Here E = C (R) and f n = ( 1) n δ (n), elements of E (R). The Borel-Ritt problem. Given a sector S : α < arg z < β, z < r. Find an analytic function φ on S such that on any subsector S 1 : α 1 < arg z < β 1 one has φ(z) a n z n, z +. (3) n= In our setting, one may consider E the space of analytic functions on S having expansions of the form (3). The f n are the linear functionals sending ϕ to its n-th coefficient of the expansion.

27 Particular cases The Borel problem: φ (n) (), n =, 1, 2,.... Here E = C (R) and f n = ( 1) n δ (n), elements of E (R). The Borel-Ritt problem. Given a sector S : α < arg z < β, z < r. Find an analytic function φ on S such that on any subsector S 1 : α 1 < arg z < β 1 one has φ(z) a n z n, z +. (3) n= In our setting, one may consider E the space of analytic functions on S having expansions of the form (3). The f n are the linear functionals sending ϕ to its n-th coefficient of the expansion.

28 Particular case: General Stieltjes moment problems for rapidly decreasing smooth functions Direct generalization of Pólya-Boas-Durán problem, x n φ(x)dx, n =, 1, 2,..., where φ S(, ). Distribution moment problem: f n, φ, n =, 1, 2,..., (4) where f n S [, ) (= f n S (R) with supp f n [, )).

29 Particular case: General Stieltjes moment problems for rapidly decreasing smooth functions Direct generalization of Pólya-Boas-Durán problem, x n φ(x)dx, n =, 1, 2,..., where φ S(, ). Distribution moment problem: f n, φ, n =, 1, 2,..., (4) where f n S [, ) (= f n S (R) with supp f n [, )).

30 Particular case: General Stieltjes moment problems for rapidly decreasing smooth functions Direct generalization of Pólya-Boas-Durán problem, x n φ(x)dx, n =, 1, 2,..., where φ S(, ). Distribution moment problem: f n, φ, n =, 1, 2,..., (4) where f n S [, ) (= f n S (R) with supp f n [, )).

31 Particular cases 1 Continuous generalized moment problem f n (x)φ(x)dx, n =, 1, 2, Discrete problem: Let (B k,n ) be an infinite matrix B k,n φ(k), n =, 1, 2,..., k=1 or, more generally, < λ n, B k,n φ(λ k ), n =, 1, 2,.... k=1 3 Let {F n } n= be a sequence of functions of local bounded variation (having at most polynomial growth) φ(x)df n (x), n =, 1, 2,....

32 Particular cases 1 Continuous generalized moment problem f n (x)φ(x)dx, n =, 1, 2, Discrete problem: Let (B k,n ) be an infinite matrix B k,n φ(k), n =, 1, 2,..., k=1 or, more generally, < λ n, B k,n φ(λ k ), n =, 1, 2,.... k=1 3 Let {F n } n= be a sequence of functions of local bounded variation (having at most polynomial growth) φ(x)df n (x), n =, 1, 2,....

33 Particular cases 1 Continuous generalized moment problem f n (x)φ(x)dx, n =, 1, 2, Discrete problem: Let (B k,n ) be an infinite matrix B k,n φ(k), n =, 1, 2,..., k=1 or, more generally, < λ n, B k,n φ(λ k ), n =, 1, 2,.... k=1 3 Let {F n } n= be a sequence of functions of local bounded variation (having at most polynomial growth) φ(x)df n (x), n =, 1, 2,....

34 Particular cases 1 Continuous generalized moment problem f n (x)φ(x)dx, n =, 1, 2, Discrete problem: Let (B k,n ) be an infinite matrix B k,n φ(k), n =, 1, 2,..., k=1 or, more generally, < λ n, B k,n φ(λ k ), n =, 1, 2,.... k=1 3 Let {F n } n= be a sequence of functions of local bounded variation (having at most polynomial growth) φ(x)df n (x), n =, 1, 2,....

35 Back to the abstract moment problem We now consider the abstract moment problem f n, φ, n =, 1, 2,..., where E is an FS-space. So, f n E and φ E. Fréchet space: locally convex, metrizable, and complete TVS. Every Fréchet space E is the projective limit of a decreasing sequence of Banach spaces E = proj lim E j E n+1 E n... E 1, with E n+1 E n continuous and dense. Fréchet-Schwartz (FS): E n+1 E n are compact.

36 Back to the abstract moment problem We now consider the abstract moment problem f n, φ, n =, 1, 2,..., where E is an FS-space. So, f n E and φ E. Fréchet space: locally convex, metrizable, and complete TVS. Every Fréchet space E is the projective limit of a decreasing sequence of Banach spaces E = proj lim E j E n+1 E n... E 1, with E n+1 E n continuous and dense. Fréchet-Schwartz (FS): E n+1 E n are compact.

37 Back to the abstract moment problem We now consider the abstract moment problem f n, φ, n =, 1, 2,..., where E is an FS-space. So, f n E and φ E. Fréchet space: locally convex, metrizable, and complete TVS. Every Fréchet space E is the projective limit of a decreasing sequence of Banach spaces E = proj lim E j E n+1 E n... E 1, with E n+1 E n continuous and dense. Fréchet-Schwartz (FS): E n+1 E n are compact.

38 Back to the abstract moment problem We now consider the abstract moment problem f n, φ, n =, 1, 2,..., where E is an FS-space. So, f n E and φ E. Fréchet space: locally convex, metrizable, and complete TVS. Every Fréchet space E is the projective limit of a decreasing sequence of Banach spaces E = proj lim E j E n+1 E n... E 1, with E n+1 E n continuous and dense. Fréchet-Schwartz (FS): E n+1 E n are compact.

39 Silva s duality theory for FS-spaces A DFS-space (or Silva space) is the inductive limit of an increasing sequence of Banach spaces, X 1 X 2... X n X n+1 ind lim X j = X, where each X n X n+1 is compact and injective. Silva s Lemma: Y X is closed if and only if Y X n is closed in X n, n. Silva s Duality Theorem: The dual of an FS-space is a DFS-space. The dual of a DFS-space is an FS-space. The FS- and DFS-spaces are Montel (hence reflexive). If E = proj lim E n, with E j+1 E j compact and dense, then E = ind lim E n.

40 Silva s duality theory for FS-spaces A DFS-space (or Silva space) is the inductive limit of an increasing sequence of Banach spaces, X 1 X 2... X n X n+1 ind lim X j = X, where each X n X n+1 is compact and injective. Silva s Lemma: Y X is closed if and only if Y X n is closed in X n, n. Silva s Duality Theorem: The dual of an FS-space is a DFS-space. The dual of a DFS-space is an FS-space. The FS- and DFS-spaces are Montel (hence reflexive). If E = proj lim E n, with E j+1 E j compact and dense, then E = ind lim E n.

41 Silva s duality theory for FS-spaces A DFS-space (or Silva space) is the inductive limit of an increasing sequence of Banach spaces, X 1 X 2... X n X n+1 ind lim X j = X, where each X n X n+1 is compact and injective. Silva s Lemma: Y X is closed if and only if Y X n is closed in X n, n. Silva s Duality Theorem: The dual of an FS-space is a DFS-space. The dual of a DFS-space is an FS-space. The FS- and DFS-spaces are Montel (hence reflexive). If E = proj lim E n, with E j+1 E j compact and dense, then E = ind lim E n.

42 Simplest examples of FS- and DFS-spaces Let P n the space of polynomials of degree n (in one variable). So, P n = C n+1. Consider the canonical injections ι n : P n P n+1 and the projections π n : P n+1 P n. The space of polynomials P = ind lim P n is DFS. The space of formal power series C[[ξ]] = proj lim P n is FS. Duality P = C[[ξ]] and (C[[ξ]]) = P.

43 Simplest examples of FS- and DFS-spaces Let P n the space of polynomials of degree n (in one variable). So, P n = C n+1. Consider the canonical injections ι n : P n P n+1 and the projections π n : P n+1 P n. The space of polynomials P = ind lim P n is DFS. The space of formal power series C[[ξ]] = proj lim P n is FS. Duality P = C[[ξ]] and (C[[ξ]]) = P.

44 Simplest examples of FS- and DFS-spaces Let P n the space of polynomials of degree n (in one variable). So, P n = C n+1. Consider the canonical injections ι n : P n P n+1 and the projections π n : P n+1 P n. The space of polynomials P = ind lim P n is DFS. The space of formal power series C[[ξ]] = proj lim P n is FS. Duality P = C[[ξ]] and (C[[ξ]]) = P.

45 Abstract moment problem in FS-spaces Theorem Let E = proj lim E j be an FS-space, where E n+1 E n is compact and dense. Consider {f n } n= E. Every arbitrary abstract moment problem has solution φ E if and only if f n, φ = a n, n N, 1 f, f 1,..., f n,..., are linearly independent. 2 span{f n : n N} E j is finite dimensional, j N. Proof. Sketch on the blackboard. We use the following lemma: Lemma. Let X = ind lim X j be a DFS-space, with X j X j+1 compact and injective. A continuous injective mapping L : P X has closed range if and only if L(P) X j is finite dimensional j.

46 Abstract moment problem in FS-spaces Theorem Let E = proj lim E j be an FS-space, where E n+1 E n is compact and dense. Consider {f n } n= E. Every arbitrary abstract moment problem has solution φ E if and only if f n, φ = a n, n N, 1 f, f 1,..., f n,..., are linearly independent. 2 span{f n : n N} E j is finite dimensional, j N. Proof. Sketch on the blackboard. We use the following lemma: Lemma. Let X = ind lim X j be a DFS-space, with X j X j+1 compact and injective. A continuous injective mapping L : P X has closed range if and only if L(P) X j is finite dimensional j.

47 Abstract moment problem in FS-spaces Theorem Let E = proj lim E j be an FS-space, where E n+1 E n is compact and dense. Consider {f n } n= E. Every arbitrary abstract moment problem has solution φ E if and only if f n, φ = a n, n N, 1 f, f 1,..., f n,..., are linearly independent. 2 span{f n : n N} E j is finite dimensional, j N. Proof. Sketch on the blackboard. We use the following lemma: Lemma. Let X = ind lim X j be a DFS-space, with X j X j+1 compact and injective. A continuous injective mapping L : P X has closed range if and only if L(P) X j is finite dimensional j.

48 Applications For the Borel problem: φ (n) () = ( 1) n δ (n), φ, n =, 1, 2,..., one takes E = C (R) = proj lim C j [ j, j]. Since all elements of the dual of C j [ j, j] are derivatives of order j + 1 of measures, the last theorem implies that every Borel problem has solution. A similar argument shows that every Borel-Ritt problem has a solution. For the Stieltjes moment problem, one writes where S p (, ) is S(, ) = proj lim S p (, ), {ψ C p (, ) : ψ (j) () = and lim x x p ψ (j) (x) =, j p}

49 Applications For the Borel problem: φ (n) () = ( 1) n δ (n), φ, n =, 1, 2,..., one takes E = C (R) = proj lim C j [ j, j]. Since all elements of the dual of C j [ j, j] are derivatives of order j + 1 of measures, the last theorem implies that every Borel problem has solution. A similar argument shows that every Borel-Ritt problem has a solution. For the Stieltjes moment problem, one writes where S p (, ) is S(, ) = proj lim S p (, ), {ψ C p (, ) : ψ (j) () = and lim x x p ψ (j) (x) =, j p}

50 Applications For the Borel problem: φ (n) () = ( 1) n δ (n), φ, n =, 1, 2,..., one takes E = C (R) = proj lim C j [ j, j]. Since all elements of the dual of C j [ j, j] are derivatives of order j + 1 of measures, the last theorem implies that every Borel problem has solution. A similar argument shows that every Borel-Ritt problem has a solution. For the Stieltjes moment problem, one writes where S p (, ) is S(, ) = proj lim S p (, ), {ψ C p (, ) : ψ (j) () = and lim x x p ψ (j) (x) =, j p}

51 Applications {x n } n= S p(, ), p. The theorem yields that x n φ(x)dx = a n, n =, 1, 2,..., has always a solution φ S(, ). More generally, it is possible to characterize the {f n } n= S[, ) for which every generalized Stieltjes moment problem f n, φ = a n, n N, admits a solution φ S(, ) in terms of the asymptotic behavior of the linear combinations of {f n } n=.

52 Applications {x n } n= S p(, ), p. The theorem yields that x n φ(x)dx = a n, n =, 1, 2,..., has always a solution φ S(, ). More generally, it is possible to characterize the {f n } n= S[, ) for which every generalized Stieltjes moment problem f n, φ = a n, n N, admits a solution φ S(, ) in terms of the asymptotic behavior of the linear combinations of {f n } n=.

53 The weighted Stieltjes moment problem Let F on [, ) and let {α n } n Z C. Theorem Every weighted Stieltjes moment problem φ(x)x αn df (x), n Z, has a solution φ S(, ), provided that: 1 The sets {n Z : M Re α n M} are finite M >. 2 If lim n Re α n =, then < lim sup x log x F(t)dt log x.

54 The weighted Stieltjes moment problem Let F on [, ) and let {α n } n Z C. Theorem Every weighted Stieltjes moment problem φ(x)x αn df (x), n Z, has a solution φ S(, ), provided that: 1 The sets {n Z : M Re α n M} are finite M >. 2 If lim n Re α n =, then < lim sup x log x F(t)dt log x.

55 The weighted Stieltjes moment problem Let F on [, ) and let {α n } n Z C. Theorem Every weighted Stieltjes moment problem φ(x)x αn df (x), n Z, has a solution φ S(, ), provided that: 1 The sets {n Z : M Re α n M} are finite M >. 2 If lim n Re α n =, then < lim sup x log x F(t)dt log x.

56 Examples The following generalized moment problems always have a solution φ S(, ). Let {α n } n= be such that Re α n. k αn φ(k), n =, 1, 2,.... k=1 p prime p αn φ(p), n =, 1, 2,....

57 Examples The following generalized moment problems always have a solution φ S(, ). Let {α n } n= be such that Re α n. k αn φ(k), n =, 1, 2,.... k=1 p prime p αn φ(p), n =, 1, 2,....