Introduction ORFs and ARFs Relations Favard theorem Orthogonal Rational Functions, Associated Rational Functions and Functions of the Second Kind Karl Deckers Department of Computer Science, Katholieke Universiteit Leuven, Heverlee, Belgium. PhD student. Supervisor: Adhemar Bultheel. July 2008 Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Outline Introduction 1 Preliminaries ORFs and ARFs 2 Orthogonal rational functions Associated rational functions Relations 3 ARFs of different order ARFs and functions of the second kind Favard theorem 4 Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Outline Introduction 1 Preliminaries ORFs and ARFs 2 Orthogonal rational functions Associated rational functions Relations 3 ARFs of different order ARFs and functions of the second kind Favard theorem 4 Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Outline Introduction 1 Preliminaries ORFs and ARFs 2 Orthogonal rational functions Associated rational functions Relations 3 ARFs of different order ARFs and functions of the second kind Favard theorem 4 Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Outline Introduction 1 Preliminaries ORFs and ARFs 2 Orthogonal rational functions Associated rational functions Relations 3 ARFs of different order ARFs and functions of the second kind Favard theorem 4 Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Outline Introduction 1 Preliminaries ORFs and ARFs 2 Orthogonal rational functions Associated rational functions Relations 3 ARFs of different order ARFs and functions of the second kind Favard theorem 4 Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Inner product Inner product consider an inner product defined by a linear functional M : � f , g � = M { fg ∗ } , g ∗ ( x ) = g ( x ) . assume M is Hermitian positive-definite and normalized (HPDN); i.e. ∀ f , g : M { fg ∗ } = M { f ∗ g } (Hermitian) ∀ f � = 0 : M { ff ∗ } > 0 (positive-definite) M { 1 } = 1 (normalized) . Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Inner product Inner product consider an inner product defined by a linear functional M : � f , g � = M { fg ∗ } , g ∗ ( x ) = g ( x ) . assume M is Hermitian positive-definite and normalized (HPDN); i.e. ∀ f , g : M { fg ∗ } = M { f ∗ g } (Hermitian) ∀ f � = 0 : M { ff ∗ } > 0 (positive-definite) M { 1 } = 1 (normalized) . Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Inner product Inner product consider an inner product defined by a linear functional M : � f , g � = M { fg ∗ } , g ∗ ( x ) = g ( x ) . assume M is Hermitian positive-definite and normalized (HPDN); i.e. ∀ f , g : M { fg ∗ } = M { f ∗ g } (Hermitian) ∀ f � = 0 : M { ff ∗ } > 0 (positive-definite) M { 1 } = 1 (normalized) . Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Inner product Inner product consider an inner product defined by a linear functional M : � f , g � = M { fg ∗ } , g ∗ ( x ) = g ( x ) . assume M is Hermitian positive-definite and normalized (HPDN); i.e. ∀ f , g : M { fg ∗ } = M { f ∗ g } (Hermitian) ∀ f � = 0 : M { ff ∗ } > 0 (positive-definite) M { 1 } = 1 (normalized) . Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Inner product Inner product consider an inner product defined by a linear functional M : � f , g � = M { fg ∗ } , g ∗ ( x ) = g ( x ) . assume M is Hermitian positive-definite and normalized (HPDN); i.e. ∀ f , g : M { fg ∗ } = M { f ∗ g } (Hermitian) ∀ f � = 0 : M { ff ∗ } > 0 (positive-definite) M { 1 } = 1 (normalized) . Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Orthogonal polynomials Orthonormal polynomials (OPs) P n = space of polynomials of degree ≤ n . canonical basis for P n : P n = span { 1 , x , x 2 , . . . , x n } . suppose a sequence of polynomials exists so that � 0 , n � = j ∀ n ≥ 0 : p n ∈ P n \ P n − 1 and M { p n p j ∗ } = 1 , n = j . Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Orthogonal polynomials Orthonormal polynomials (OPs) P n = space of polynomials of degree ≤ n . canonical basis for P n : P n = span { 1 , x , x 2 , . . . , x n } . suppose a sequence of polynomials exists so that � 0 , n � = j ∀ n ≥ 0 : p n ∈ P n \ P n − 1 and M { p n p j ∗ } = 1 , n = j . Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Orthogonal polynomials Orthonormal polynomials (OPs) P n = space of polynomials of degree ≤ n . canonical basis for P n : P n = span { 1 , x , x 2 , . . . , x n } . suppose a sequence of polynomials exists so that � 0 , n � = j ∀ n ≥ 0 : p n ∈ P n \ P n − 1 and M { p n p j ∗ } = 1 , n = j . Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Orthogonal polynomials Orthonormal polynomials (OPs) These OPs satisfy the following 3-term recurrence relation: p − 1 ( x ) ≡ 0 , p 0 ( x ) ≡ 1 , p n ( x ) = E n ( x + F n ) p n − 1 ( x ) + C n p n − 2 ( x ) , C n = − E n n > 0 , E n � = 0 , E n − 1 � = 0 . Associated polynomials (APs) APs p ( k ) n − k ∈ P n − k of order k are defined by: p ( k ) p ( k ) − 1 ( x ) ≡ 0 , 0 ( x ) ≡ 1 , p ( k ) n − k ( x ) = E n ( x + F n ) p ( k ) n − 1 − k ( x ) + C n p ( k ) n − 2 − k ( x ) , n > k . Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Orthogonal polynomials Orthonormal polynomials (OPs) These OPs satisfy the following 3-term recurrence relation: p − 1 ( x ) ≡ 0 , p 0 ( x ) ≡ 1 , p n ( x ) = E n ( x + F n ) p n − 1 ( x ) + C n p n − 2 ( x ) , C n = − E n n > 0 , E n � = 0 , E n − 1 � = 0 . Associated polynomials (APs) APs p ( k ) n − k ∈ P n − k of order k are defined by: p ( k ) p ( k ) − 1 ( x ) ≡ 0 , 0 ( x ) ≡ 1 , p ( k ) n − k ( x ) = E n ( x + F n ) p ( k ) n − 1 − k ( x ) + C n p ( k ) n − 2 − k ( x ) , n > k . Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Associated polynomials Associated polynomials (APs) Note that APs of order 0 = OPs, ∀ n ≥ k ≥ 0 : p ( k ) n − k ∈ P n − k \ P n − k − 1 ⇒ Favard theorem: there exists a HPDN linear functional M ( k ) so that � 0 , n � = j M ( k ) { p ( k ) n − k p ( k ) j − k ∗ } = 1 , n = j . Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Associated polynomials Associated polynomials (APs) Note that APs of order 0 = OPs, ∀ n ≥ k ≥ 0 : p ( k ) n − k ∈ P n − k \ P n − k − 1 ⇒ Favard theorem: there exists a HPDN linear functional M ( k ) so that � 0 , n � = j M ( k ) { p ( k ) n − k p ( k ) j − k ∗ } = 1 , n = j . Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Associated polynomials Associated polynomials (APs) Note that APs of order 0 = OPs, ∀ n ≥ k ≥ 0 : p ( k ) n − k ∈ P n − k \ P n − k − 1 ⇒ Favard theorem: there exists a HPDN linear functional M ( k ) so that � 0 , n � = j M ( k ) { p ( k ) n − k p ( k ) j − k ∗ } = 1 , n = j . Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Associated polynomials Relation APs of different order (R1) p ( k ) n − k ( x ) = E k +1 ( x + F k +1 ) p ( k +1) n − ( k +1) ( x ) + C k +2 p ( k +2) n − ( k +2) ( x ), n > k (R2) p ( k ) n − k ( x ) = p ( j ) n − j ( x ) p ( k ) j − k ( x ) + C j +1 p ( j +1) n − ( j +1) ( x ) p ( k ) ( j − 1) − k ( x ), k + 1 ≤ j ≤ n − 1 APs and functions of the second kind p ( k − 1) n − ( k − 1) ( t ) − p ( k − 1) � � n − ( k − 1) ( x ) (R3) p ( k ) E k · M ( k − 1) 1 n − k ( x ) = t t − x Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Associated polynomials Relation APs of different order (R1) p ( k ) n − k ( x ) = E k +1 ( x + F k +1 ) p ( k +1) n − ( k +1) ( x ) + C k +2 p ( k +2) n − ( k +2) ( x ), n > k (R2) p ( k ) n − k ( x ) = p ( j ) n − j ( x ) p ( k ) j − k ( x ) + C j +1 p ( j +1) n − ( j +1) ( x ) p ( k ) ( j − 1) − k ( x ), k + 1 ≤ j ≤ n − 1 APs and functions of the second kind p ( k − 1) n − ( k − 1) ( t ) − p ( k − 1) � � n − ( k − 1) ( x ) (R3) p ( k ) E k · M ( k − 1) 1 n − k ( x ) = t t − x Karl Deckers ORFs and ARFs
Introduction ORFs and ARFs Relations Favard theorem Associated polynomials Relation APs of different order (R1) p ( k ) n − k ( x ) = E k +1 ( x + F k +1 ) p ( k +1) n − ( k +1) ( x ) + C k +2 p ( k +2) n − ( k +2) ( x ), n > k (R2) p ( k ) n − k ( x ) = p ( j ) n − j ( x ) p ( k ) j − k ( x ) + C j +1 p ( j +1) n − ( j +1) ( x ) p ( k ) ( j − 1) − k ( x ), k + 1 ≤ j ≤ n − 1 APs and functions of the second kind p ( k − 1) n − ( k − 1) ( t ) − p ( k − 1) � � n − ( k − 1) ( x ) (R3) p ( k ) E k · M ( k − 1) 1 n − k ( x ) = t t − x Karl Deckers ORFs and ARFs
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