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Holomorphic functions associated with indeterminate rational moment problems Adhemar Bultheel 1 , Erik Hendriksen 2 , Olav Nj astad 3 1 Dept. Computer Science, KU Leuven 2 Nieuwkoop, The Netherlands 3 Mathematics, Univ. Trondheim Puerto de la

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Holomorphic functions associated with indeterminate rational moment problems

Adhemar Bultheel1, Erik Hendriksen2, Olav Nj˚ astad3

  • 1Dept. Computer Science, KU Leuven

2Nieuwkoop, The Netherlands 3Mathematics, Univ. Trondheim

Puerto de la Cruz, Tenerife, January 2014 This is dedicated to the memory of Pablo Gonz´ alez Vera.

http://nalag.cs.kuleuven.be/papers/ade/growth2

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 1 / 15

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Survey

Rational Hamburger moment problem ∞ ⇒ {1/αk} Solution µ ⇔ Nevanlinna functions Ωµ(z) All solutions µ ⇔ Ωµ(z) = A(z)g(z)+B(z)

C(z)g(z)+D(z), g ∈ N

Asymptotic behaviour of A, B,C, D near singularities ({αk}) is the same

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 2 / 15

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Classical Hamburger moment problem

Definition (Hamburger MP)

Given M HPD functional on polynomials P by M[tk] = ck, k = 0, 1, ... Find pos. measure µ on R such that

  • tkdµ(t) = ck, k = 0, 1, ...

Set of all solutions = M. MP is (in)determinate if solution is (not) unique.

Definition (Nevanlinna function)

N = {f : f ∈ H(U) & f : U → ˆ U = U ∪ R}. Example Sµ(z) = dµ(t)

t−z for µ ∈ M.

Expand (t − z)−1 to think of Sµ as a moment generating fct: Sµ(z) ∼ −1 z

∞

  • k=0

ck zk .

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 3 / 15

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Classical Hamburger moment problem

Definition (Hamburger MP)

Given M HPD functional on polynomials P by M[tk] = ck, k = 0, 1, ... Find pos. measure µ on R such that

  • tkdµ(t) = ck, k = 0, 1, ...

Set of all solutions = M. MP is (in)determinate if solution is (not) unique.

Definition (Nevanlinna function)

N = {f : f ∈ H(U) & f : U → ˆ U = U ∪ R}. Example Sµ(z) = dµ(t)

t−z for µ ∈ M.

Expand (t − z)−1 to think of Sµ as a moment generating fct: Sµ(z) ∼ −1 z

∞

  • k=0

ck zk .

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 3 / 15

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Classical Hamburger moment problem

Definition (Hamburger MP)

Given M HPD functional on polynomials P by M[tk] = ck, k = 0, 1, ... Find pos. measure µ on R such that

  • tkdµ(t) = ck, k = 0, 1, ...

Set of all solutions = M. MP is (in)determinate if solution is (not) unique.

Definition (Nevanlinna function)

N = {f : f ∈ H(U) & f : U → ˆ U = U ∪ R}. Example Sµ(z) = dµ(t)

t−z for µ ∈ M.

Expand (t − z)−1 to think of Sµ as a moment generating fct: Sµ(z) ∼ −1 z

∞

  • k=0

ck zk .

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 3 / 15

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Classical Hamburger moment problem

Theorem (Nevanlinna)

There is a 1-1 relation between N and M given by Sµ(z) = − a(z)f (z) − c(z) b(z)f (z) − d(z), f ∈ N with a, b, c, d entire functions.

Theorem (M. Riesz)

If F ∈ {a, b, c, d} then for any ǫ > 0 |F(z)| ≤ M(ǫ) exp{ǫ|z|}. (controls growth as z → ∞)

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 4 / 15

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Classical Hamburger moment problem

Theorem (Nevanlinna)

There is a 1-1 relation between N and M given by Sµ(z) = − a(z)f (z) − c(z) b(z)f (z) − d(z), f ∈ N with a, b, c, d entire functions.

Theorem (M. Riesz)

If F ∈ {a, b, c, d} then for any ǫ > 0 |F(z)| ≤ M(ǫ) exp{ǫ|z|}. (controls growth as z → ∞)

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 4 / 15

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Rational moment problem

rational case

How to generalize this to the case of rational moments? I.e. when polynomials P is replaced by rationals L L = P π∞ , with π∞(z) =

  • α∈A

(1 − z α) Here a finite set Γ of different α’s in ˆ R \ {0} = (R ∪ {∞}) \ {0}. But each α ∈ Γ is has infinite multiplicity. Then L · L = L If Γ = {∞} then L = P. Here for notational reason α = ∞.

This presentation

We shall look in particular at the growth of the a, b, c, d near the singularities α ∈ Γ.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 5 / 15

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Rational moment problem

rational case

How to generalize this to the case of rational moments? I.e. when polynomials P is replaced by rationals L L = P π∞ , with π∞(z) =

  • α∈A

(1 − z α) Here a finite set Γ of different α’s in ˆ R \ {0} = (R ∪ {∞}) \ {0}. But each α ∈ Γ is has infinite multiplicity. Then L · L = L If Γ = {∞} then L = P. Here for notational reason α = ∞.

This presentation

We shall look in particular at the growth of the a, b, c, d near the singularities α ∈ Γ.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 5 / 15

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SLIDE 10

Rational moment problem

rational case

How to generalize this to the case of rational moments? I.e. when polynomials P is replaced by rationals L L = P π∞ , with π∞(z) =

  • α∈A

(1 − z α) Here a finite set Γ of different α’s in ˆ R \ {0} = (R ∪ {∞}) \ {0}. But each α ∈ Γ is has infinite multiplicity. Then L · L = L If Γ = {∞} then L = P. Here for notational reason α = ∞.

This presentation

We shall look in particular at the growth of the a, b, c, d near the singularities α ∈ Γ.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 5 / 15

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Rational moment problem

rational case

How to generalize this to the case of rational moments? I.e. when polynomials P is replaced by rationals L L = P π∞ , with π∞(z) =

  • α∈A

(1 − z α) Here a finite set Γ of different α’s in ˆ R \ {0} = (R ∪ {∞}) \ {0}. But each α ∈ Γ is has infinite multiplicity. Then L · L = L If Γ = {∞} then L = P. Here for notational reason α = ∞.

This presentation

We shall look in particular at the growth of the a, b, c, d near the singularities α ∈ Γ.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 5 / 15

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Rational moment problem

rational case

How to generalize this to the case of rational moments? I.e. when polynomials P is replaced by rationals L L = P π∞ , with π∞(z) =

  • α∈A

(1 − z α) Here a finite set Γ of different α’s in ˆ R \ {0} = (R ∪ {∞}) \ {0}. But each α ∈ Γ is has infinite multiplicity. Then L · L = L If Γ = {∞} then L = P. Here for notational reason α = ∞.

This presentation

We shall look in particular at the growth of the a, b, c, d near the singularities α ∈ Γ.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 5 / 15

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Rational moment problem

rational case

How to generalize this to the case of rational moments? I.e. when polynomials P is replaced by rationals L L = P π∞ , with π∞(z) =

  • α∈A

(1 − z α) Here a finite set Γ of different α’s in ˆ R \ {0} = (R ∪ {∞}) \ {0}. But each α ∈ Γ is has infinite multiplicity. Then L · L = L If Γ = {∞} then L = P. Here for notational reason α = ∞.

This presentation

We shall look in particular at the growth of the a, b, c, d near the singularities α ∈ Γ.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 5 / 15

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Orthogonal rational functions

wlog: {α0 = ∞, α1, . . . , αq

  • Γ

, αq+1 = α1, . . . , α2q = αq

  • Γ

, . . .} π0 = 1, πn = n

k=1(1 − z αk ),

bn(z) =

zn πn(z), n = 1, 2, . . .

Ln = span{bk(z), k = 0, ...n} = pn(z) πn(z), pn ∈ Pn

  • Definition (Rational moment problem)

Given M HPD functional on L by ck = M[bk], k = 0, 1, .... Find pos. measure µ such that ck =

  • bk(t)dµ(t), k = 0, 1, ...

Definition (Nevanlinna function)

D(t, z) = 1 + tz t − z ⇒ Ωµ(z) =

  • D(t, z)dµ(t) ∈ N

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 6 / 15

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Orthogonal rational functions

wlog: {α0 = ∞, α1, . . . , αq

  • Γ

, αq+1 = α1, . . . , α2q = αq

  • Γ

, . . .} π0 = 1, πn = n

k=1(1 − z αk ),

bn(z) =

zn πn(z), n = 1, 2, . . .

Ln = span{bk(z), k = 0, ...n} = pn(z) πn(z), pn ∈ Pn

  • Definition (Rational moment problem)

Given M HPD functional on L by ck = M[bk], k = 0, 1, .... Find pos. measure µ such that ck =

  • bk(t)dµ(t), k = 0, 1, ...

Definition (Nevanlinna function)

D(t, z) = 1 + tz t − z ⇒ Ωµ(z) =

  • D(t, z)dµ(t) ∈ N

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 6 / 15

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Orthogonal rational functions

wlog: {α0 = ∞, α1, . . . , αq

  • Γ

, αq+1 = α1, . . . , α2q = αq

  • Γ

, . . .} π0 = 1, πn = n

k=1(1 − z αk ),

bn(z) =

zn πn(z), n = 1, 2, . . .

Ln = span{bk(z), k = 0, ...n} = pn(z) πn(z), pn ∈ Pn

  • Definition (Rational moment problem)

Given M HPD functional on L by ck = M[bk], k = 0, 1, .... Find pos. measure µ such that ck =

  • bk(t)dµ(t), k = 0, 1, ...

Definition (Nevanlinna function)

D(t, z) = 1 + tz t − z ⇒ Ωµ(z) =

  • D(t, z)dµ(t) ∈ N

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 6 / 15

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Orthogonal rational functions

wlog: {α0 = ∞, α1, . . . , αq

  • Γ

, αq+1 = α1, . . . , α2q = αq

  • Γ

, . . .} π0 = 1, πn = n

k=1(1 − z αk ),

bn(z) =

zn πn(z), n = 1, 2, . . .

Ln = span{bk(z), k = 0, ...n} = pn(z) πn(z), pn ∈ Pn

  • Definition (Rational moment problem)

Given M HPD functional on L by ck = M[bk], k = 0, 1, .... Find pos. measure µ such that ck =

  • bk(t)dµ(t), k = 0, 1, ...

Definition (Nevanlinna function)

D(t, z) = 1 + tz t − z ⇒ Ωµ(z) =

  • D(t, z)dµ(t) ∈ N

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 6 / 15

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Orthogonal rational functions

f , g = M[f (z)g(z)] =

  • f (t)g(t)dµ(t), µ ∈ M

⇒ ORF: ϕn ∈ Ln \ Ln−1, ϕk, ϕl = δk,l functions of second kind: ψ0(z) = −z and ψn(z) =

  • D(t, z)[ϕn(t) − ϕn(z)]dµ(t),

n ≥ 1, µ ∈ M. ... quadrature formulas, Christoffel-Darboux, (long story) ...

Theorem

RMP indeterminate (determinate) iff

∞

  • n=0

|ϕn(z)|2 (and

∞

  • n=0

|ψn(z)|2) converge (diverge) for all C \ R.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 7 / 15

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Orthogonal rational functions

f , g = M[f (z)g(z)] =

  • f (t)g(t)dµ(t), µ ∈ M

⇒ ORF: ϕn ∈ Ln \ Ln−1, ϕk, ϕl = δk,l functions of second kind: ψ0(z) = −z and ψn(z) =

  • D(t, z)[ϕn(t) − ϕn(z)]dµ(t),

n ≥ 1, µ ∈ M. ... quadrature formulas, Christoffel-Darboux, (long story) ...

Theorem

RMP indeterminate (determinate) iff

∞

  • n=0

|ϕn(z)|2 (and

∞

  • n=0

|ψn(z)|2) converge (diverge) for all C \ R.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 7 / 15

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Orthogonal rational functions

f , g = M[f (z)g(z)] =

  • f (t)g(t)dµ(t), µ ∈ M

⇒ ORF: ϕn ∈ Ln \ Ln−1, ϕk, ϕl = δk,l functions of second kind: ψ0(z) = −z and ψn(z) =

  • D(t, z)[ϕn(t) − ϕn(z)]dµ(t),

n ≥ 1, µ ∈ M. ... quadrature formulas, Christoffel-Darboux, (long story) ...

Theorem

RMP indeterminate (determinate) iff

∞

  • n=0

|ϕn(z)|2 (and

∞

  • n=0

|ψn(z)|2) converge (diverge) for all C \ R.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 7 / 15

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SLIDE 21

Orthogonal rational functions

f , g = M[f (z)g(z)] =

  • f (t)g(t)dµ(t), µ ∈ M

⇒ ORF: ϕn ∈ Ln \ Ln−1, ϕk, ϕl = δk,l functions of second kind: ψ0(z) = −z and ψn(z) =

  • D(t, z)[ϕn(t) − ϕn(z)]dµ(t),

n ≥ 1, µ ∈ M. ... quadrature formulas, Christoffel-Darboux, (long story) ...

Theorem

RMP indeterminate (determinate) iff

∞

  • n=0

|ϕn(z)|2 (and

∞

  • n=0

|ψn(z)|2) converge (diverge) for all C \ R.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 7 / 15

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Rational Nevanlinna parametrization

Lemma

There exists x0 ∈ R \ (Γ ∪ {0}) not ‘exceptional’ such that An(z) Cn(z) Bn(z) Dn(z)

  • = (x0 − z)

n−1

  • k=0

Mk(z) ∈ L2×2

n

M0 =

  • −1

D(z, x0) −D(z, x0) 1

  • , Mk =

ψk(z) ϕk(z)

  • [ψk(x0) ϕk(x0)], k ≥ 1

and the limit n → ∞ exists locally uniformly in C \ Γ giving A, B, C, D holomorphic with a simple pole at ∞ and essential singularities at the points of Γ = {α1, . . . , αq}.

  • dµ(t) < ∞ ⇒ simple pole at ∞
  • Γ-points repeated ∞ times ⇒ essential singularities

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 8 / 15

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Rational Nevanlinna parametrization

Lemma

There exists x0 ∈ R \ (Γ ∪ {0}) not ‘exceptional’ such that An(z) Cn(z) Bn(z) Dn(z)

  • = (x0 − z)

n−1

  • k=0

Mk(z) ∈ L2×2

n

M0 =

  • −1

D(z, x0) −D(z, x0) 1

  • , Mk =

ψk(z) ϕk(z)

  • [ψk(x0) ϕk(x0)], k ≥ 1

and the limit n → ∞ exists locally uniformly in C \ Γ giving A, B, C, D holomorphic with a simple pole at ∞ and essential singularities at the points of Γ = {α1, . . . , αq}.

  • dµ(t) < ∞ ⇒ simple pole at ∞
  • Γ-points repeated ∞ times ⇒ essential singularities

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 8 / 15

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Rational Nevanlinna parametrization

Lemma

There exists x0 ∈ R \ (Γ ∪ {0}) not ‘exceptional’ such that An(z) Cn(z) Bn(z) Dn(z)

  • = (x0 − z)

n−1

  • k=0

Mk(z) ∈ L2×2

n

M0 =

  • −1

D(z, x0) −D(z, x0) 1

  • , Mk =

ψk(z) ϕk(z)

  • [ψk(x0) ϕk(x0)], k ≥ 1

and the limit n → ∞ exists locally uniformly in C \ Γ giving A, B, C, D holomorphic with a simple pole at ∞ and essential singularities at the points of Γ = {α1, . . . , αq}.

  • dµ(t) < ∞ ⇒ simple pole at ∞
  • Γ-points repeated ∞ times ⇒ essential singularities

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 8 / 15

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Rational Nevanlinna parametrization

Theorem (Nevanlinna parametrization)

Ωµ(z) = − A(z)g(z) − C(z) B(z)g(z) − D(z) is a 1-1 relation between all g ∈ N and all µ ∈ M.

Theorem (Rational Riesz theorem)

If F ∈ {A, B, C, D} then for any ǫ > 0 near a singularity α ∈ Γ |F(z)| ≤ M(ǫ) exp

  • ǫ

|z − α|

  • ∀z ∈ V ′

α (a pointed disk centered at α excluding all other α’s)

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 9 / 15

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Rational Nevanlinna parametrization

Theorem (Nevanlinna parametrization)

Ωµ(z) = − A(z)g(z) − C(z) B(z)g(z) − D(z) is a 1-1 relation between all g ∈ N and all µ ∈ M.

Theorem (Rational Riesz theorem)

If F ∈ {A, B, C, D} then for any ǫ > 0 near a singularity α ∈ Γ |F(z)| ≤ M(ǫ) exp

  • ǫ

|z − α|

  • ∀z ∈ V ′

α (a pointed disk centered at α excluding all other α’s)

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 9 / 15

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Order and type

Definition (order and type at α ∈ Γ)

Mα(F, r) = max

|z−α|=r |F(z)|,

then order is defined by ρα(F) = inf{λ : Mα(F, r) ≤ exp{r−λ}, r small} and type is defined by σα(F) = inf{s : Mα(F, r) ≤ exp{sr−ρα(F)}, r small}

Theorem

For F ∈ {A, B, C, D} and α ∈ Γ either (1) ρα(F) < 1 or (2) ρα(F) = 1 and σα(F) = 0.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 10 / 15

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SLIDE 28

Order and type

Definition (order and type at α ∈ Γ)

Mα(F, r) = max

|z−α|=r |F(z)|,

then order is defined by ρα(F) = inf{λ : Mα(F, r) ≤ exp{r−λ}, r small} and type is defined by σα(F) = inf{s : Mα(F, r) ≤ exp{sr−ρα(F)}, r small}

Theorem

For F ∈ {A, B, C, D} and α ∈ Γ either (1) ρα(F) < 1 or (2) ρα(F) = 1 and σα(F) = 0.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 10 / 15

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Two solutions

Define 2 solutions: µ0 and µ∞ corresponding to g = ∞ and g = 0 A(z) B(z) = −Ωµ∞(z) and C(z) D(z) = −Ωµ0(z), for z ∈ ˆ C \ Γ supp µ∞ = Γ ∪ poles of A/B; zeros of A and B simple and interlace supp µ0 = Γ ∪ poles of C/D; zeros of C and D simple and interlace thus supp µ∞ = Γ ∪ zeros of B & supp µ0 = Γ ∪ zeros of D But also the zeros of B and D interlace. Hence µ∞ and µ0 have common support Γ but all other points in the support interlace.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 11 / 15

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SLIDE 30

Two solutions

Define 2 solutions: µ0 and µ∞ corresponding to g = ∞ and g = 0 A(z) B(z) = −Ωµ∞(z) and C(z) D(z) = −Ωµ0(z), for z ∈ ˆ C \ Γ supp µ∞ = Γ ∪ poles of A/B; zeros of A and B simple and interlace supp µ0 = Γ ∪ poles of C/D; zeros of C and D simple and interlace thus supp µ∞ = Γ ∪ zeros of B & supp µ0 = Γ ∪ zeros of D But also the zeros of B and D interlace. Hence µ∞ and µ0 have common support Γ but all other points in the support interlace.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 11 / 15

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SLIDE 31

Two solutions

Define 2 solutions: µ0 and µ∞ corresponding to g = ∞ and g = 0 A(z) B(z) = −Ωµ∞(z) and C(z) D(z) = −Ωµ0(z), for z ∈ ˆ C \ Γ supp µ∞ = Γ ∪ poles of A/B; zeros of A and B simple and interlace supp µ0 = Γ ∪ poles of C/D; zeros of C and D simple and interlace thus supp µ∞ = Γ ∪ zeros of B & supp µ0 = Γ ∪ zeros of D But also the zeros of B and D interlace. Hence µ∞ and µ0 have common support Γ but all other points in the support interlace.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 11 / 15

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Two solutions

Define 2 solutions: µ0 and µ∞ corresponding to g = ∞ and g = 0 A(z) B(z) = −Ωµ∞(z) and C(z) D(z) = −Ωµ0(z), for z ∈ ˆ C \ Γ supp µ∞ = Γ ∪ poles of A/B; zeros of A and B simple and interlace supp µ0 = Γ ∪ poles of C/D; zeros of C and D simple and interlace thus supp µ∞ = Γ ∪ zeros of B & supp µ0 = Γ ∪ zeros of D But also the zeros of B and D interlace. Hence µ∞ and µ0 have common support Γ but all other points in the support interlace.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 11 / 15

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SLIDE 33

Two solutions

Define 2 solutions: µ0 and µ∞ corresponding to g = ∞ and g = 0 A(z) B(z) = −Ωµ∞(z) and C(z) D(z) = −Ωµ0(z), for z ∈ ˆ C \ Γ supp µ∞ = Γ ∪ poles of A/B; zeros of A and B simple and interlace supp µ0 = Γ ∪ poles of C/D; zeros of C and D simple and interlace thus supp µ∞ = Γ ∪ zeros of B & supp µ0 = Γ ∪ zeros of D But also the zeros of B and D interlace. Hence µ∞ and µ0 have common support Γ but all other points in the support interlace.

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 11 / 15

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SLIDE 34

Accumulation points

Each α ∈ Γ is an accumulation point in supp µ∞ and in supp µ0. Define {zF

j : j = 1, 2, ...} zeros of F and

{zF

α,j : j = 1, 2, ...} (disjoint) subsequence that converges to α ∈ Γ.

The latter exist for F ∈ {B, D} because {zB

j } ⊂ supp µ∞ and {zD j } ⊂ supp µ0

but also for F ∈ {A, C} by interlacing property. How do the {zF

α,j} converge to α ∈ Γ?

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 12 / 15

slide-35
SLIDE 35

Accumulation points

Each α ∈ Γ is an accumulation point in supp µ∞ and in supp µ0. Define {zF

j : j = 1, 2, ...} zeros of F and

{zF

α,j : j = 1, 2, ...} (disjoint) subsequence that converges to α ∈ Γ.

The latter exist for F ∈ {B, D} because {zB

j } ⊂ supp µ∞ and {zD j } ⊂ supp µ0

but also for F ∈ {A, C} by interlacing property. How do the {zF

α,j} converge to α ∈ Γ?

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 12 / 15

slide-36
SLIDE 36

Accumulation points

Each α ∈ Γ is an accumulation point in supp µ∞ and in supp µ0. Define {zF

j : j = 1, 2, ...} zeros of F and

{zF

α,j : j = 1, 2, ...} (disjoint) subsequence that converges to α ∈ Γ.

The latter exist for F ∈ {B, D} because {zB

j } ⊂ supp µ∞ and {zD j } ⊂ supp µ0

but also for F ∈ {A, C} by interlacing property. How do the {zF

α,j} converge to α ∈ Γ?

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 12 / 15

slide-37
SLIDE 37

Accumulation points

Each α ∈ Γ is an accumulation point in supp µ∞ and in supp µ0. Define {zF

j : j = 1, 2, ...} zeros of F and

{zF

α,j : j = 1, 2, ...} (disjoint) subsequence that converges to α ∈ Γ.

The latter exist for F ∈ {B, D} because {zB

j } ⊂ supp µ∞ and {zD j } ⊂ supp µ0

but also for F ∈ {A, C} by interlacing property. How do the {zF

α,j} converge to α ∈ Γ?

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 12 / 15

slide-38
SLIDE 38

Accumulation points

Definition (cvg exp and genus at α ∈ Γ)

  • rder {zF

α,j} such that |zF α,j − α| non-increasing

then convergence exponent is defined by τα(F) = inf{t ∈ R : ∞

j=1 |zF α,j − α|t < ∞}

and genus is defined by κα(F) = max{t ∈ Z : ∞

j=1 |zF α,j − α|t = ∞}

Theorem

For α ∈ Γ: τα(A) = τα(B) = τα(C) = τα(D) and κα(A) = κα(B) = κα(C) = κα(D).

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 13 / 15

slide-39
SLIDE 39

Accumulation points

Definition (cvg exp and genus at α ∈ Γ)

  • rder {zF

α,j} such that |zF α,j − α| non-increasing

then convergence exponent is defined by τα(F) = inf{t ∈ R : ∞

j=1 |zF α,j − α|t < ∞}

and genus is defined by κα(F) = max{t ∈ Z : ∞

j=1 |zF α,j − α|t = ∞}

Theorem

For α ∈ Γ: τα(A) = τα(B) = τα(C) = τα(D) and κα(A) = κα(B) = κα(C) = κα(D).

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 13 / 15

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SLIDE 40

Factorization

Theorem

F ∈ {A, B, C, D} then F(z) = R(z)

α∈Γ Fα(z), Fα(z) = Pα(z)Qα(z).

R(z) rational with zeros and poles in Γ. Qα ∈ H(ˆ C \ {α}) without zeros: Qα = eq(

1 z−α ), ∂q ≤ 1

Pα ∈ H(ˆ C \ {α}) is canonical Hadamard product I.e., catches all the zeros {zF

α,j}.

Pα(z) =

∞

  • j=1

 

  • 1 −

zF

α,j − α

z − α

  • exp

  

γα(F)

  • k=1

1 k

  • zF

α,j − α

z − α k     γα(F) = ⌊ρα(F)⌋ here γα(F) ∈ {0, 1}

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 14 / 15

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SLIDE 41

Equality of orders

For α ∈ Γ: ρα(F) completely defined by Fα: ρα(F) = ρα(Fα) = τα(Fα) Eventually leads to

Theorem (Equality of orders)

For α ∈ Γ it holds that ρα(A) = ρα(B) = ρα(C) = ρα(D)

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 15 / 15

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SLIDE 42

Equality of orders

For α ∈ Γ: ρα(F) completely defined by Fα: ρα(F) = ρα(Fα) = τα(Fα) Eventually leads to

Theorem (Equality of orders)

For α ∈ Γ it holds that ρα(A) = ρα(B) = ρα(C) = ρα(D)

Bultheel, Hendriksen, Nj˚ astad (KU Leuven) Indeterminate rational moment problem Tenerife January 2014 15 / 15