AIMS-VOLKSWAGEN STIFTUNG WORKSHOP, DOUALA, 2018. Classical discrete d -orthogonal polynomials Naoures AYADI nawres.essths@gmail.com Under the direction of Hamza CHAGGARA Lab. Mathematics-Physics, Special Functions and Applications ESSTHS, SOUSSE UNIVERSITY, TUNISIA October 11, 2018
Plan 1 Naoures AYADI | Classical discrete d -orthogonal polynomials
Plan 1 1 Preliminaries and Notations. Naoures AYADI | Classical discrete d -orthogonal polynomials
Plan 1 1 Preliminaries and Notations. 2 Orthogonality and Classical discrete OPSs Naoures AYADI | Classical discrete d -orthogonal polynomials
Plan 1 1 Preliminaries and Notations. 2 Orthogonality and Classical discrete OPSs 3 d -Orthogonality and Classical discrete d -OPSs Naoures AYADI | Classical discrete d -orthogonal polynomials
Plan 1 1 Preliminaries and Notations. 2 Orthogonality and Classical discrete OPSs 3 d -Orthogonality and Classical discrete d -OPSs 4 Examples Naoures AYADI | Classical discrete d -orthogonal polynomials
Plan 1 1 Preliminaries and Notations. 2 Orthogonality and Classical discrete OPSs 3 d -Orthogonality and Classical discrete d -OPSs 4 Examples • ∆ ω -Appell polynomial set: d -OPS of Charlier type Naoures AYADI | Classical discrete d -orthogonal polynomials
Plan 1 1 Preliminaries and Notations. 2 Orthogonality and Classical discrete OPSs 3 d -Orthogonality and Classical discrete d -OPSs 4 Examples • ∆ ω -Appell polynomial set: d -OPS of Charlier type • 2-OPS of Meixner type Naoures AYADI | Classical discrete d -orthogonal polynomials
Preliminaries and Notations 2 • P : The vector space of polynomials with coefficients in C and P ′ its dual. Naoures AYADI | Classical discrete d -orthogonal polynomials
Preliminaries and Notations 2 • P : The vector space of polynomials with coefficients in C and P ′ its dual. • PS: The polynomial set { P n } n � 0 ∈ P , such that deg P n = n for all n . Naoures AYADI | Classical discrete d -orthogonal polynomials
Preliminaries and Notations 2 • P : The vector space of polynomials with coefficients in C and P ′ its dual. • PS: The polynomial set { P n } n � 0 ∈ P , such that deg P n = n for all n . • �L , f � : The duality brackets between P ′ and P ( L ∈ P ′ and f ∈ P ). Naoures AYADI | Classical discrete d -orthogonal polynomials
Preliminaries and Notations 2 • P : The vector space of polynomials with coefficients in C and P ′ its dual. • PS: The polynomial set { P n } n � 0 ∈ P , such that deg P n = n for all n . • �L , f � : The duality brackets between P ′ and P ( L ∈ P ′ and f ∈ P ). • A PS, { P n } n � 0 , is monic if P n = x n + a n − 1 x n − 1 + · · · + a 0 , n � 0. Naoures AYADI | Classical discrete d -orthogonal polynomials
Preliminaries and Notations 2 • P : The vector space of polynomials with coefficients in C and P ′ its dual. • PS: The polynomial set { P n } n � 0 ∈ P , such that deg P n = n for all n . • �L , f � : The duality brackets between P ′ and P ( L ∈ P ′ and f ∈ P ). • A PS, { P n } n � 0 , is monic if P n = x n + a n − 1 x n − 1 + · · · + a 0 , n � 0. • τ b : The translation operator ( τ b f )( x ) = f ( x − b ) , f ∈ P , b ∈ R . Naoures AYADI | Classical discrete d -orthogonal polynomials
Preliminaries and Notations 2 • P : The vector space of polynomials with coefficients in C and P ′ its dual. • PS: The polynomial set { P n } n � 0 ∈ P , such that deg P n = n for all n . • �L , f � : The duality brackets between P ′ and P ( L ∈ P ′ and f ∈ P ). • A PS, { P n } n � 0 , is monic if P n = x n + a n − 1 x n − 1 + · · · + a 0 , n � 0. • τ b : The translation operator ( τ b f )( x ) = f ( x − b ) , f ∈ P , b ∈ R . • ∆ ω , ∇ ω : Hahn’s operators ∆ ω ( f )( x ) = f ( x + ω ) − f ( x ) , and ω ∇ ω f ( x ) = f ( x ) − f ( x − ω ) = ∆ − ω f ( x ); ω � = 0 , f ∈ P . ω Naoures AYADI | Classical discrete d -orthogonal polynomials
Preliminaries and Notations 2 • P : The vector space of polynomials with coefficients in C and P ′ its dual. • PS: The polynomial set { P n } n � 0 ∈ P , such that deg P n = n for all n . • �L , f � : The duality brackets between P ′ and P ( L ∈ P ′ and f ∈ P ). • A PS, { P n } n � 0 , is monic if P n = x n + a n − 1 x n − 1 + · · · + a 0 , n � 0. • τ b : The translation operator ( τ b f )( x ) = f ( x − b ) , f ∈ P , b ∈ R . • ∆ ω , ∇ ω : Hahn’s operators ∆ ω ( f )( x ) = f ( x + ω ) − f ( x ) , and ω ∇ ω f ( x ) = f ( x ) − f ( x − ω ) = ∆ − ω f ( x ); ω � = 0 , f ∈ P . ω • ∆ 1 ≡ ∆ and ∇ 1 ≡ ∇ . Naoures AYADI | Classical discrete d -orthogonal polynomials
Preliminaries and Notations 2 • P : The vector space of polynomials with coefficients in C and P ′ its dual. • PS: The polynomial set { P n } n � 0 ∈ P , such that deg P n = n for all n . • �L , f � : The duality brackets between P ′ and P ( L ∈ P ′ and f ∈ P ). • A PS, { P n } n � 0 , is monic if P n = x n + a n − 1 x n − 1 + · · · + a 0 , n � 0. • τ b : The translation operator ( τ b f )( x ) = f ( x − b ) , f ∈ P , b ∈ R . • ∆ ω , ∇ ω : Hahn’s operators ∆ ω ( f )( x ) = f ( x + ω ) − f ( x ) , and ω ∇ ω f ( x ) = f ( x ) − f ( x − ω ) = ∆ − ω f ( x ); ω � = 0 , f ∈ P . ω • ∆ 1 ≡ ∆ and ∇ 1 ≡ ∇ . • �L , ∆ ω f � = −�∇ ω L , f � , L ∈ P ′ , f ∈ P . Naoures AYADI | Classical discrete d -orthogonal polynomials
Preliminaries and Notations 2 • P : The vector space of polynomials with coefficients in C and P ′ its dual. • PS: The polynomial set { P n } n � 0 ∈ P , such that deg P n = n for all n . • �L , f � : The duality brackets between P ′ and P ( L ∈ P ′ and f ∈ P ). • A PS, { P n } n � 0 , is monic if P n = x n + a n − 1 x n − 1 + · · · + a 0 , n � 0. • τ b : The translation operator ( τ b f )( x ) = f ( x − b ) , f ∈ P , b ∈ R . • ∆ ω , ∇ ω : Hahn’s operators ∆ ω ( f )( x ) = f ( x + ω ) − f ( x ) , and ω ∇ ω f ( x ) = f ( x ) − f ( x − ω ) = ∆ − ω f ( x ); ω � = 0 , f ∈ P . ω • ∆ 1 ≡ ∆ and ∇ 1 ≡ ∇ . • �L , ∆ ω f � = −�∇ ω L , f � , L ∈ P ′ , f ∈ P . • ( µ n ) n := � u , x n � , n � 0: The moment sequence of u ∈ P ′ . Naoures AYADI | Classical discrete d -orthogonal polynomials
Preliminaries and Notations 3 Definition Let { P n } n � 0 be a monic PS. We call dual sequence of { P n } n � 0 , the sequence of linear forms {F n } n � 0 defined by �F n , P m � = δ n , m ; n , m � 0. Naoures AYADI | Classical discrete d -orthogonal polynomials
Preliminaries and Notations 3 Definition Let { P n } n � 0 be a monic PS. We call dual sequence of { P n } n � 0 , the sequence of linear forms {F n } n � 0 defined by �F n , P m � = δ n , m ; n , m � 0. Lemma Let L ∈ P ′ , k � 1, and { P n } n � 0 ∈ P . In order that L satisfies �L , P k − 1 � � = 0 , and �L , P n � = 0 ; n � k , (1) it is necessary and sufficient that there exists λ ν ∈ C , 0 � ν � k − 1 , λ k − 1 � = 0, such that k − 1 � L = λ ν F ν . (2) ν = 0 Naoures AYADI | Classical discrete d -orthogonal polynomials
Orthogonality 4 Definition A PS, { P n } n � 0 , is said to be an orthogonal polynomial sequence (OPS) with respect to a functional L if: �L , P n P m � = K n δ n , m ( K n � = 0 ) , (3) for n , m = 0 , 1 , 2 , · · · . Naoures AYADI | Classical discrete d -orthogonal polynomials
Orthogonality 4 Definition A PS, { P n } n � 0 , is said to be an orthogonal polynomial sequence (OPS) with respect to a functional L if: �L , P n P m � = K n δ n , m ( K n � = 0 ) , (3) for n , m = 0 , 1 , 2 , · · · . � The functional L is called regular. Naoures AYADI | Classical discrete d -orthogonal polynomials
Orthogonality 5 Definition Orthogonal polynomials of discrete variables are polynomials P n , n = 0 , 1 ... that satisfy the orthogonality relation � P n ( x i ) P m ( x i ) ρ ( x i ) = 0 , n � = m , (4) i where ρ is a positive weight function. The summation is done on the values of x which satisfy a � x i < b where x i + 1 = x i + 1, a and b are finite or infinite. Naoures AYADI | Classical discrete d -orthogonal polynomials
∆ ω -Classical OPSs 6 Lemma Let { P n } n � 0 be a monic PS and {F n } n � 0 its dual sequence. We consider { Q ω n } n � 0 defined by n ( x ) = ∆ ω P n + 1 ( x ) Q ω , n � 0 , (5) n + 1 � � � F n and n � 0 its associated dual sequence, then we have ∇ ω � F n = − ( n + 1 ) F n + 1 , n � 0 . (6) Naoures AYADI | Classical discrete d -orthogonal polynomials
∆ ω -Classical OPSs 6 Lemma Let { P n } n � 0 be a monic PS and {F n } n � 0 its dual sequence. We consider { Q ω n } n � 0 defined by n ( x ) = ∆ ω P n + 1 ( x ) Q ω , n � 0 , (5) n + 1 � � � F n and n � 0 its associated dual sequence, then we have ∇ ω � F n = − ( n + 1 ) F n + 1 , n � 0 . (6) Definition A PS { P n } n � 0 is said to be classical discrete orthogonal polynomial or ∆ ω -Classical OPS if { Q ω n } n � 0 is also OPS. Naoures AYADI | Classical discrete d -orthogonal polynomials
Recommend
More recommend
