Orthogonal rational functions and rational modifications of a - PowerPoint PPT Presentation

Introduction Preliminaries Measures on T Examples Measures on I Orthogonal rational functions and rational modifications of a measure Karl Deckers Department of Computer Science, Katholieke Universiteit Leuven, Heverlee, Belgium. PhD student. Supervisor: Adhemar Bultheel. September 30, 2008 Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Outline Introduction 1 Preliminaries 2 Measures on T 3 Examples 4 Measures on I 5 Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Motivation Explicit examples of orthogonal rational functions (ORFs) Chebyshev ORFs on I = [ − 1 , 1] w.r.t. � ± 1 � w ( x ) = (1 − x ) a (1 + x ) b , a , b ∈ . 2 Takenaka-Malmquist basis on T = { z : | z | 2 = 1 } w.r.t. Lebesgue measure Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Motivation Relating I and T θ ∈ [ − π, π ] , z = e i θ w ( θ ) = w (cos θ ) | sin θ | , ˚ µ ′ ( z ) ( a , b ) w ( x ) w ( θ ) ˚ ˚ 1 − x 2 � − 1 / 2 − 1 2 , − 1 1 � � � 1 2 i z 1 − x 2 � 1 / 2 sin 2 θ - ( z 2 − 1) 2 � 1 2 , 1 � � 2 4 i z 3 � 1 � 1 / 2 � - ( z − 1) 2 2 , − 1 1 − x � 1 − cos θ 2 1+ x 2 i z 2 � 1 / 2 ( z +1) 2 � − 1 2 , 1 1+ x � � 1 + cos θ 2 i z 2 2 1 − x µ ) → φ n ( z ; ˚ � φ n ( x ; w ) Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Orthogonal polynomials Polynomial modifications µ = | z ± 1 | 2 d ˚ d ˜ µ = (1 ± cos θ ) d ˚ µ z = e i θ , sin 2 θ d ˚ µ = | z 2 − 1 | 2 d ˚ d ˆ µ = µ | p m ( z ) | 2 d ˚ d ˜ µ = µ | ( z − γ 1 ) · . . . · ( z − γ m ) | 2 d ˚ = µ, | γ j | ≤ 1 , j = 1 , . . . , m φ n ( z ; ˜ µ ) ⊥ ˜ µ P n − 1 ⇒ p m ( z ) φ n ( z ; ˜ µ ) ⊥ ˚ µ p m ( z ) P n − 1 ⇒ relation φ n ( z ; ˜ µ ) and φ n + m ( z ; ˚ µ ) Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Orthogonal rational functions Rational modifications | r m ( z ) | 2 d ˚ d ˜ µ = µ 2 � � ( z − γ 1 ) · ... · ( z − γ m ) = d ˚ µ, � � (1 − α 1 z ) · ... · (1 − α m z ) � � | γ j | ≤ 1 , and | α j | < 1 , j = 1 , . . . , m µ ˚ µ r m ( z ) ˚ φ n ( z ; ˜ µ ) ⊥ ˜ L n − 1 ⇒ r m ( z ) φ n ( z ; ˜ µ ) ⊥ ˚ L n − 1 ⇒ relation φ n ( z ; ˜ µ ) and φ n + m ( z ; ˚ µ ) Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Canonical basis for ˚ L Blaschke factors and Blaschke products canonical basis for ˚ L n : � β k | β k | , β k � = 0 z − β k ζ β k ( z ) = η β k 1 − β k z , η β k = 1 , β k = 0 B k ( z ) = ζ β k ( z ) B k − 1 ( z ) , B 0 ( z ) ≡ 1 canonical basis for ˜ L m : ˜ B k ( z ) = ζ α k ( z )˜ B k − 1 ( z ) canonical basis for ˆ L n + m : � ˜ B k ( z ) , k ≤ m ˆ B k ( z ) = ˜ B m ( z ) B k − m ( z ) , k > m Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Definitions Monic ORFs φ n ( z ) is called monic iff φ n ( z ) = 1 · B n ( z ) + f n − 1 ( z ), f n − 1 ∈ ˚ L n − 1 or equivalently φ ∗ n ( β n ) = 1 where φ ∗ n ( z ) = B n ( z ) φ n ∗ ( z ) and φ n ∗ ( z ) = φ n (1 / z ) . Suppose φ n ( z ) = a · B n ( z ) + f n − 1 ( z ), then φ ∗ n ( z ) = B n ( z ) ( a · B n ∗ ( z ) + f n − 1 ∗ ( z )) a + ζ β n ( z ) f ∗ B k ∗ ( z ) = B − 1 � � = n − 1 ( z ) , k ( z ) . Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Definitions From now on we assume φ k ’s are monic ϕ k ’s are orthonormal, with ϕ k = ˚ κ k φ k Reproducing kernel ˚ � n k n ( z , u ; ˚ µ ) = k =0 ϕ k ( z ; ˚ µ ) ϕ k ( u ; ˚ µ ) κ n +1 | 2 φ ∗ n +1 ( z ;˚ µ ) φ ∗ n +1 ( u ;˚ µ ) − φ n +1 ( z ;˚ µ ) φ n +1 ( u ;˚ µ ) = | ˚ , n > 0 1 − ζ β n +1 ( z ) ζ β n +1 ( u ) Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Rational modification Relating monic ORFs µ ˚ µ ˆ φ n ( z ; ˜ µ ) ⊥ ˜ L n − 1 and φ n + m ( z ; ˚ µ ) ⊥ ˚ L n + m − 1 µ ) − r m ( z ) µ ) ∈ ˆ φ n + m ( z ; ˚ m ( β n ) φ n ( z ; ˜ L n + m − 1 r ∗ because φ ∗ n + m ( β n ; ˚ µ ) = 1 and µ )] ∗ ˆ [ r m ( z ) φ n ( z ; ˜ = B n + m ( z ) r m ∗ ( z ) φ n ∗ ( z ; ˜ µ ) ˜ µ ) = r ∗ m ( z ) φ ∗ = B m ( z ) r m ∗ ( z ) B n ( z ) φ n ∗ ( z ; ˜ n ( z ; ˜ µ ) . Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Rational modification Basis for ˆ L n + m − 1 Consider orthogonal decomposition � ⊥ ˚ � µ L n + m − 1 = r m ˚ ˆ r m ˚ L n − 1 ⊕ L n − 1 µ ) } n − 1 k =0 is orthogonal basis for r m ˚ { r m φ k ( z ; ˜ L n − 1 w.r.t. ˚ µ µ where � j � ⊥ ˚ � � { g i , k ( z ) } m i − 1 r m ˚ i =1 is basis for L n − 1 k =0 � g i , k ( z ) = ∂ k ˆ k n + m − 1 ( z , u ; ˚ µ ) � � ∂ u k � � u = γ i Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Rational modification µ ) − r m ( z ) φ n + m ( z ; ˚ m ( β n ) φ n ( z ; ˜ µ ) = r ∗ j m i − 1 n − 1 � � � Λ k r m ( z ) φ k ( z ; ˜ µ ) + λ i , k g i , k ( z ) i =1 k =0 k =0 φ n + m ( z ; ˚ µ ) ⊥ ˚ µ r m ( z ) φ k ( z ; ˜ µ ) and r m ( z ) φ n ( z ; ˜ µ ) ⊥ ˚ µ r m ( z ) φ k ( z ; ˜ µ ), so that Λ k = 0. j m i − 1 µ ) − r m ( z ) � � ⇒ φ n + m ( z ; ˚ m ( β n ) φ n ( z ; ˜ µ ) = λ i , k g i , k ( z ) r ∗ i =1 k =0 Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Rational modification Suppose the zeros γ i are simple: m µ ) − r m ( z ) λ j ˆ � φ n + m ( z ; ˚ m ( β n ) φ n ( z ; ˜ µ ) = k n + m − 1 ( z , γ j ; ˚ µ ) . r ∗ j =1 m � λ j ˆ φ n + m ( γ i ; ˚ µ ) = k n + m − 1 ( γ i , γ j ; ˚ µ ) , i = 1 , . . . , m . j =1 λ = K − 1 φ n + m (˚ µ ) . Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Rational modification Theorem µ where r m ∈ ˜ L m \ ˜ µ = | r m ( z ) | 2 d ˚ Let d ˜ L m − 1 with simple zeros in µ ) denote the monic ORF in ˚ { γ j } m j =1 . Let φ n ( z ; ˜ L n w.r.t. ˜ µ . µ ) denote the monic ORF in ˆ Similarly, let φ n + m ( z ; ˚ L n + m w.r.t. ˚ µ . Then � T � ˆ r m ( z ) 1 φ n + m ( z ; ˚ µ ) k m + n − 1 ( z ; ˚ µ ) � � m ( β n ) φ n ( z ; ˜ µ ) = � � r ∗ det K � φ n + m (˚ µ ) K � � � Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Computational aspects Computing the monic ORFs computing φ n ( z ; ˚ µ m ) by means of intermediate results; i.e. rational modifications of degree 1: � � z − γ � � d ˚ µ 1 = � d ˚ µ � � 1 − α z � � z − γ � φ n ( z ; ˚ µ 1 ) 1 − α z � � 1 − γβ n µ ) − φ n +1 ( γ ; ˚ µ ) ˆ = η α φ n +1 ( z ; ˚ k n ( z , γ ; ˚ µ ) . ˆ 1 − αβ n k n ( γ, γ ; ˚ µ ) distinction between | γ | = 1 and | γ | < 1 Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Computational aspects | γ | = 1 1 − γβ n η α ( z − γ ) 2 1 − αβ n 1 − α z φ n ( z ; ˚ µ 1 ) = × φ ∗ � � φ n +1 ( γ ; ˚ µ ) n +1 ( γ ; ˚ µ ) � � � ′ φ ∗ ′ � φ n +1 ( γ ; ˚ µ ) n +1 ( γ ; ˚ µ ) � � φ ∗ � � ( z − γ ) φ n +1 ( z ; ˚ µ ) φ n +1 ( z ; ˚ µ ) n +1 ( z ; ˚ µ ) � � � φ ∗ � 0 φ n +1 ( γ ; ˚ µ ) n +1 ( γ ; ˚ µ ) , � � � � � � 1 − β n z ′ φ ∗ ′ φ n +1 ( γ ; ˚ µ ) φ n +1 ( γ ; ˚ µ ) n +1 ( γ ; ˚ µ ) � � � 1 − β n γ � ′ represents the derivative of φ . where φ Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Computational aspects | γ | < 1 1 − γβ n η α (1 − γ z )( z − γ ) 1 − αβ n φ n ( z ; ˚ µ 1 ) = × φ ∗ 1 − α z � � φ n +1 ( γ ; ˚ µ ) n +1 ( γ ; ˚ µ ) � � � φ ∗ � n +1 ( γ ; ˚ µ ) φ n +1 ( γ ; ˚ µ ) � � c n ( z ) φ ∗ � � (1 − γ z ) φ n +1 ( z ; ˚ µ ) c n ( z ) φ n +1 ( z ; ˚ µ ) n +1 ( z ; ˚ µ ) � � � � 1 −| γ | 2 � � φ ∗ φ n +1 ( γ ; ˚ µ ) φ n +1 ( γ ; ˚ µ ) n +1 ( γ ; ˚ µ ) , � � 1 − β n γ � � � � φ ∗ 0 n +1 ( γ ; ˚ µ ) φ n +1 ( γ ; ˚ µ ) � � where c n ( z ) = (1 − β n z ). Karl Deckers ORFs and RM

Introduction Preliminaries Measures on T Examples Measures on I Examples Chebyshev ORFs on T ˚ w ( θ ) = 1 ± cos θ a n + z ( z − b n ) zB n − 1 ( z ) 1 − β n z φ n ( z ; ˚ w ) = c n ( z ± 1) 2 w ( θ ) = sin 2 θ ˚ d n + e n z + z 2 ( f n + g n z + z 2 ) zB n − 1 ( z ) 1 − β n z φ n ( z ; ˚ w ) = h n ( z 2 − 1) 2 Karl Deckers ORFs and RM