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 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 Indeterminate rational moment problem Tenerife January 2014 1 / 15
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Holomorphic functions associated with indeterminaterational moment problems
Adhemar Bultheel1, Erik Hendriksen2, Olav Njastad3
1Dept. Computer Science, KU Leuven2Nieuwkoop, The Netherlands
3Mathematics, Univ. Trondheim
Puerto de la Cruz, Tenerife, January 2014
This is dedicated to the memory of Pablo Gonzalez Vera.
http://nalag.cs.kuleuven.be/papers/ade/growth2
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 1 / 15
Survey
Rational Hamburger moment problem ∞⇒ 1/αkSolution 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 thesame
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 2 / 15
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→ 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
ckzk.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 3 / 15
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→ 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
ckzk.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 3 / 15
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→ 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
ckzk.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 3 / 15
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 ε > 0
|F (z)| ≤ M(ε) expε|z |.
(controls growth as z →∞)
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 4 / 15
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 ε > 0
|F (z)| ≤ M(ε) expε|z |.
(controls growth as z →∞)
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 4 / 15
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 =Pπ∞
, with π∞(z) =∏α∈A
(1− z
α)
Here a finite set Γ of different α’s in R \ 0 = (R ∪ ∞) \ 0.But each α ∈ Γ is has infinite multiplicity. Then L · L = LIf Γ = ∞ then L = P. Here for notational reason α 6=∞.
This presentation
We shall look in particular at the growth of the a, b, c , d near thesingularities α ∈ Γ.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 5 / 15
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 =Pπ∞
, with π∞(z) =∏α∈A
(1− z
α)
Here a finite set Γ of different α’s in R \ 0 = (R ∪ ∞) \ 0.But each α ∈ Γ is has infinite multiplicity. Then L · L = LIf Γ = ∞ then L = P. Here for notational reason α 6=∞.
This presentation
We shall look in particular at the growth of the a, b, c , d near thesingularities α ∈ Γ.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 5 / 15
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 =Pπ∞
, with π∞(z) =∏α∈A
(1− z
α)
Here a finite set Γ of different α’s in R \ 0 = (R ∪ ∞) \ 0.But each α ∈ Γ is has infinite multiplicity. Then L · L = LIf Γ = ∞ then L = P. Here for notational reason α 6=∞.
This presentation
We shall look in particular at the growth of the a, b, c , d near thesingularities α ∈ Γ.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 5 / 15
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 =Pπ∞
, with π∞(z) =∏α∈A
(1− z
α)
Here a finite set Γ of different α’s in R \ 0 = (R ∪ ∞) \ 0.But each α ∈ Γ is has infinite multiplicity. Then L · L = LIf Γ = ∞ then L = P. Here for notational reason α 6=∞.
This presentation
We shall look in particular at the growth of the a, b, c , d near thesingularities α ∈ Γ.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 5 / 15
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 =Pπ∞
, with π∞(z) =∏α∈A
(1− z
α)
Here a finite set Γ of different α’s in R \ 0 = (R ∪ ∞) \ 0.But each α ∈ Γ is has infinite multiplicity. Then L · L = LIf Γ = ∞ then L = P. Here for notational reason α 6=∞.
This presentation
We shall look in particular at the growth of the a, b, c , d near thesingularities α ∈ Γ.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 5 / 15
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 =Pπ∞
, with π∞(z) =∏α∈A
(1− z
α)
Here a finite set Γ of different α’s in R \ 0 = (R ∪ ∞) \ 0.But each α ∈ Γ is has infinite multiplicity. Then L · L = LIf Γ = ∞ then L = P. Here for notational reason α 6=∞.
This presentation
We shall look in particular at the growth of the a, b, c , d near thesingularities α ∈ Γ.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 5 / 15
Orthogonal rational functions
wlog: α0 =∞, α1, . . . , αq︸ ︷︷ ︸Γ
, αq+1 = α1, . . . , α2q = αq︸ ︷︷ ︸Γ
, . . .
π0 = 1, πn =∏n
k=1(1− zαk
), bn(z) = zn
πn(z) , n = 1, 2, . . .
Ln = spanbk(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, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 6 / 15
Orthogonal rational functions
wlog: α0 =∞, α1, . . . , αq︸ ︷︷ ︸Γ
, αq+1 = α1, . . . , α2q = αq︸ ︷︷ ︸Γ
, . . .
π0 = 1, πn =∏n
k=1(1− zαk
), bn(z) = zn
πn(z) , n = 1, 2, . . .
Ln = spanbk(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, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 6 / 15
Orthogonal rational functions
wlog: α0 =∞, α1, . . . , αq︸ ︷︷ ︸Γ
, αq+1 = α1, . . . , α2q = αq︸ ︷︷ ︸Γ
, . . .
π0 = 1, πn =∏n
k=1(1− zαk
), bn(z) = zn
πn(z) , n = 1, 2, . . .
Ln = spanbk(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, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 6 / 15
Orthogonal rational functions
wlog: α0 =∞, α1, . . . , αq︸ ︷︷ ︸Γ
, αq+1 = α1, . . . , α2q = αq︸ ︷︷ ︸Γ
, . . .
π0 = 1, πn =∏n
k=1(1− zαk
), bn(z) = zn
πn(z) , n = 1, 2, . . .
Ln = spanbk(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, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 6 / 15
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 7 / 15
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) ∈ L2×2n
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 ,Dholomorphic with a simple pole at ∞ and essential singularities at thepoints of Γ = α1, . . . , αq.
•∫dµ(t) <∞⇒ simple pole at ∞
• Γ-points repeated ∞ times ⇒ essential singularities
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 8 / 15
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) ∈ L2×2n
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 ,Dholomorphic with a simple pole at ∞ and essential singularities at thepoints of Γ = α1, . . . , αq.
•∫dµ(t) <∞⇒ simple pole at ∞
• Γ-points repeated ∞ times ⇒ essential singularities
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 8 / 15
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) ∈ L2×2n
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 ,Dholomorphic with a simple pole at ∞ and essential singularities at thepoints of Γ = α1, . . . , αq.
•∫dµ(t) <∞⇒ simple pole at ∞
• Γ-points repeated ∞ times ⇒ essential singularities
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 8 / 15
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 ε > 0 near a singularity α ∈ Γ
|F (z)| ≤ M(ε) exp
ε
|z − α|
∀z ∈ V ′α (a pointed disk centered at α excluding all other α’s)
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 9 / 15
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 ε > 0 near a singularity α ∈ Γ
|F (z)| ≤ M(ε) exp
ε
|z − α|
∀z ∈ V ′α (a pointed disk centered at α excluding all other α’s)
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 9 / 15
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 ) = infλ : Mα(F , r) ≤ expr−λ, r small
and type is defined by
σα(F ) = infs : Mα(F , r) ≤ expsr−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, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 10 / 15
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 ) = infλ : Mα(F , r) ≤ expr−λ, r small
and type is defined by
σα(F ) = infs : Mα(F , r) ≤ expsr−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, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 10 / 15
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 ∈ C \ Γ
suppµ∞ = Γ ∪ poles of A/B; zeros of A and B simple and interlacesuppµ0 = Γ ∪ poles of C/D; zeros of C and D simple and interlace
thussuppµ∞ = Γ ∪ zeros of B & suppµ0 = Γ ∪ zeros of D
But also the zeros of B and D interlace.
Hence µ∞ and µ0 have common support Γ butall other points in the support interlace.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 11 / 15
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 ∈ C \ Γ
suppµ∞ = Γ ∪ poles of A/B; zeros of A and B simple and interlacesuppµ0 = Γ ∪ poles of C/D; zeros of C and D simple and interlace
thussuppµ∞ = Γ ∪ zeros of B & suppµ0 = Γ ∪ zeros of D
But also the zeros of B and D interlace.
Hence µ∞ and µ0 have common support Γ butall other points in the support interlace.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 11 / 15
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 ∈ C \ Γ
suppµ∞ = Γ ∪ poles of A/B; zeros of A and B simple and interlacesuppµ0 = Γ ∪ poles of C/D; zeros of C and D simple and interlace
thussuppµ∞ = Γ ∪ zeros of B & suppµ0 = Γ ∪ zeros of D
But also the zeros of B and D interlace.
Hence µ∞ and µ0 have common support Γ butall other points in the support interlace.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 11 / 15
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 ∈ C \ Γ
suppµ∞ = Γ ∪ poles of A/B; zeros of A and B simple and interlacesuppµ0 = Γ ∪ poles of C/D; zeros of C and D simple and interlace
thussuppµ∞ = Γ ∪ zeros of B & suppµ0 = Γ ∪ zeros of D
But also the zeros of B and D interlace.
Hence µ∞ and µ0 have common support Γ butall other points in the support interlace.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 11 / 15
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 ∈ C \ Γ
suppµ∞ = Γ ∪ poles of A/B; zeros of A and B simple and interlacesuppµ0 = Γ ∪ poles of C/D; zeros of C and D simple and interlace
thussuppµ∞ = Γ ∪ zeros of B & suppµ0 = Γ ∪ zeros of D
But also the zeros of B and D interlace.
Hence µ∞ and µ0 have common support Γ butall other points in the support interlace.
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 11 / 15
Accumulation points
Each α ∈ Γ is an accumulation point in suppµ∞ and in suppµ0.
Define zFj : j = 1, 2, ... zeros of F and
zFα,j : j = 1, 2, ... (disjoint) subsequence that converges to α ∈ Γ.
The latter exist for F ∈ B,D becausezBj ⊂ suppµ∞ and zDj ⊂ suppµ0
but also for F ∈ A,C by interlacing property.
How do the zFα,j converge to α ∈ Γ?
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 12 / 15
Accumulation points
Each α ∈ Γ is an accumulation point in suppµ∞ and in suppµ0.
Define zFj : j = 1, 2, ... zeros of F and
zFα,j : j = 1, 2, ... (disjoint) subsequence that converges to α ∈ Γ.
The latter exist for F ∈ B,D becausezBj ⊂ suppµ∞ and zDj ⊂ suppµ0
but also for F ∈ A,C by interlacing property.
How do the zFα,j converge to α ∈ Γ?
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 12 / 15
Accumulation points
Each α ∈ Γ is an accumulation point in suppµ∞ and in suppµ0.
Define zFj : j = 1, 2, ... zeros of F and
zFα,j : j = 1, 2, ... (disjoint) subsequence that converges to α ∈ Γ.
The latter exist for F ∈ B,D becausezBj ⊂ suppµ∞ and zDj ⊂ suppµ0
but also for F ∈ A,C by interlacing property.
How do the zFα,j converge to α ∈ Γ?
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 12 / 15
Accumulation points
Each α ∈ Γ is an accumulation point in suppµ∞ and in suppµ0.
Define zFj : j = 1, 2, ... zeros of F and
zFα,j : j = 1, 2, ... (disjoint) subsequence that converges to α ∈ Γ.
The latter exist for F ∈ B,D becausezBj ⊂ suppµ∞ and zDj ⊂ suppµ0
but also for F ∈ A,C by interlacing property.
How do the zFα,j converge to α ∈ Γ?
Bultheel, Hendriksen, Njastad (KU Leuven)Indeterminate rational moment problem Tenerife January 2014 12 / 15
Accumulation points
Definition (cvg exp and genus at α ∈ Γ)
order zFα,j such that |zFα,j − α| non-increasingthen convergence exponent is defined by