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 (KU Leuven) Tenerife January 2014 1 / 15
Survey Rational Hamburger moment problem ∞ ⇒ { 1 /α k } Solution µ ⇔ Nevanlinna functions Ω µ ( z ) All solutions µ ⇔ Ω µ ( z ) = A ( z ) g ( z )+ B ( z ) C ( z ) g ( z )+ D ( z ) , g ∈ N Asymptotic behaviour of A , B ,C , D near singularities ( { α k } ) is the same Bultheel, Hendriksen, Nj˚ astad Indeterminate rational moment problem (KU Leuven) Tenerife January 2014 2 / 15
Classical Hamburger moment problem Definition (Hamburger MP) Given M HPD functional on polynomials P by M [ t k ] = c k , k = 0 , 1 , ... � t k d µ ( t ) = c k , k = 0 , 1 , ... Find pos. measure µ on R such that 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 } . � d µ ( t ) Example S µ ( z ) = t − z for µ ∈ M . Expand ( t − z ) − 1 to think of S µ as a moment generating fct: ∞ S µ ( z ) ∼ − 1 c k � z k . z k =0 Bultheel, Hendriksen, Nj˚ astad Indeterminate rational moment problem (KU Leuven) Tenerife January 2014 3 / 15
Classical Hamburger moment problem Definition (Hamburger MP) Given M HPD functional on polynomials P by M [ t k ] = c k , k = 0 , 1 , ... � t k d µ ( t ) = c k , k = 0 , 1 , ... Find pos. measure µ on R such that 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 } . � d µ ( t ) Example S µ ( z ) = t − z for µ ∈ M . Expand ( t − z ) − 1 to think of S µ as a moment generating fct: ∞ S µ ( z ) ∼ − 1 c k � z k . z k =0 Bultheel, Hendriksen, Nj˚ astad Indeterminate rational moment problem (KU Leuven) Tenerife January 2014 3 / 15
Classical Hamburger moment problem Definition (Hamburger MP) Given M HPD functional on polynomials P by M [ t k ] = c k , k = 0 , 1 , ... � t k d µ ( t ) = c k , k = 0 , 1 , ... Find pos. measure µ on R such that 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 } . � d µ ( t ) Example S µ ( z ) = t − z for µ ∈ M . Expand ( t − z ) − 1 to think of S µ as a moment generating fct: ∞ S µ ( z ) ∼ − 1 c k � z k . z k =0 Bultheel, Hendriksen, Nj˚ astad Indeterminate rational moment problem (KU Leuven) Tenerife January 2014 3 / 15
Classical Hamburger moment problem Theorem (Nevanlinna) There is a 1-1 relation between N and M given by S µ ( 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 Indeterminate rational moment problem (KU Leuven) Tenerife January 2014 4 / 15
Classical Hamburger moment problem Theorem (Nevanlinna) There is a 1-1 relation between N and M given by S µ ( 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 Indeterminate rational moment problem (KU Leuven) 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 = P (1 − z � , with π ∞ ( z ) = α ) π ∞ α ∈A 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 Indeterminate rational moment problem (KU Leuven) 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 = P (1 − z � , with π ∞ ( z ) = α ) π ∞ α ∈A 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 Indeterminate rational moment problem (KU Leuven) 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 = P (1 − z � , with π ∞ ( z ) = α ) π ∞ α ∈A 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 Indeterminate rational moment problem (KU Leuven) 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 = P (1 − z � , with π ∞ ( z ) = α ) π ∞ α ∈A 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 Indeterminate rational moment problem (KU Leuven) 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 = P (1 − z � , with π ∞ ( z ) = α ) π ∞ α ∈A 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 Indeterminate rational moment problem (KU Leuven) 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 = P (1 − z � , with π ∞ ( z ) = α ) π ∞ α ∈A 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 Indeterminate rational moment problem (KU Leuven) Tenerife January 2014 5 / 15
Orthogonal rational functions wlog: { α 0 = ∞ , α 1 , . . . , α q , α q +1 = α 1 , . . . , α 2 q = α q , . . . } � �� � � �� � Γ Γ π 0 = 1, π n = � n z n k =1 (1 − z α k ) , b n ( z ) = π n ( z ) , n = 1 , 2 , . . . � p n ( z ) � L n = span { b k ( z ) , k = 0 , ... n } = π n ( z ) , p n ∈ P n Definition (Rational moment problem) Given M HPD functional on L by c k = M [ b k ] , k = 0 , 1 , ... . � Find pos. measure µ such that c k = b k ( t ) d µ ( t ) , k = 0 , 1 , ... Definition (Nevanlinna function) � D ( t , z ) = 1 + tz t − z ⇒ Ω µ ( z ) = D ( t , z ) d µ ( t ) ∈ N Bultheel, Hendriksen, Nj˚ astad Indeterminate rational moment problem (KU Leuven) Tenerife January 2014 6 / 15
Orthogonal rational functions wlog: { α 0 = ∞ , α 1 , . . . , α q , α q +1 = α 1 , . . . , α 2 q = α q , . . . } � �� � � �� � Γ Γ π 0 = 1, π n = � n z n k =1 (1 − z α k ) , b n ( z ) = π n ( z ) , n = 1 , 2 , . . . � p n ( z ) � L n = span { b k ( z ) , k = 0 , ... n } = π n ( z ) , p n ∈ P n Definition (Rational moment problem) Given M HPD functional on L by c k = M [ b k ] , k = 0 , 1 , ... . � Find pos. measure µ such that c k = b k ( t ) d µ ( t ) , k = 0 , 1 , ... Definition (Nevanlinna function) � D ( t , z ) = 1 + tz t − z ⇒ Ω µ ( z ) = D ( t , z ) d µ ( t ) ∈ N Bultheel, Hendriksen, Nj˚ astad Indeterminate rational moment problem (KU Leuven) Tenerife January 2014 6 / 15
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