Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Generalized Fourier Series for Solutions of Linear Differential Equations Alexandre Benoit, Joint work with Bruno Salvy INRIA (France) February 15, 2011 1 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion I Introduction 2 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Generalized Fourier Series � f ( x ) = a n ψ n ( x ) Some Examples � ∞ ( − 1) n J 2 n ( x ) sin ( x ) = 2 n =0 � ∞ arccos ( x ) = 1 4 2 π T 0 ( x ) − T 2 n +1 ( x ) (2 n + 1) 2 π n =0 � � n � n + 1 � � � ∞ � − 1 1 � 2 erf ( x ) = 2 √ π (2 n + 1) n ! 1 F 1 � − x 4 2 n + 2 n =0 More generally ( ψ n ( x )) n ∈ N can be an orthogonal basis of a Hilbert space. 3 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Applications: Good approximation properties. Approximation of arctan(2 x ) by Taylor expansion of degree 1 1 0 . 5 x − 0 . 5 − 0 . 25 0 0 . 25 0 . 5 − 0 . 5 arctan(2 x ) Taylor approximation − 1 4 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Applications: Good approximation properties. Bad approximation outside its circle of convergence 1 . 5 1 0 . 5 x − 1 − 0 . 5 0 0 . 5 1 − 0 . 5 arctan(2 x ) − 1 Taylor approximation − 1 . 5 4 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Applications: Good approximation properties. approximation of arctan(2 x ) by Chebyshev expansion of degree 1 1 . 5 1 0 . 5 x − 1 − 0 . 5 0 0 . 5 1 − 0 . 5 arctan(2 x ) − 1 Taylor expansion − 1 . 5 Chebyshev expansion 4 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Applications: Good approximation properties. bad approximation over R 1 . 5 1 0 . 5 x − 2 − 1 0 1 2 − 0 . 5 arctan(2 x ) − 1 Taylor expansion − 1 . 5 Chebyshev expansion 4 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Applications: Good approximation properties. approximation of arctan(2 x ) by Hermite expansion of degree 1 1 . 5 1 0 . 5 x − 2 − 1 0 1 2 − 0 . 5 arctan(2 x ) − 1 Taylor expansion Chebyshev expansion − 1 . 5 Hermite expansion 4 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Our framework Families of functions ψ n ( x ) with two special properties Mult by x ( P x ) R ec x 2 ( x ψ n ( x )) = R ec x 1 ( ψ n ( x )) Examples Monomial polynomials ( M n = x n ) All orthogonal polynomials Bessel functions Legendre functions Parabolic cylinder functions 5 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Our framework Families of functions ψ n ( x ) with two special properties Mult by x ( P x ) R ec x 2 ( x ψ n ( x )) = R ec x 1 ( ψ n ( x )) Examples Monomial polynomials ( M n = x n ) xM n = M n +1 2 xT n ( x ) = T n +1 ( x ) + T n − 1 ( x ) All orthogonal polynomials 1 Bessel functions n ( x J n +1 − x J n − 1 ) = 2J n Legendre functions Parabolic cylinder functions 5 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Our framework Families of functions ψ n ( x ) with two special properties Mult by x ( P x ) R ec x 2 ( x ψ n ( x )) = R ec x 1 ( ψ n ( x )) Differentiation ( P ∂ ) R ec ∂ 2 ( ψ ′ n ( x )) = R ec ∂ 1 ( ψ n ( x )) Examples Monomial polynomials Classical orthogonal polynomials Bessel functions Legendre functions Parabolic cylinder functions 5 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Our framework Families of functions ψ n ( x ) with two special properties Mult by x ( P x ) R ec x 2 ( x ψ n ( x )) = R ec x 1 ( ψ n ( x )) Differentiation ( P ∂ ) R ec ∂ 2 ( ψ ′ n ( x )) = R ec ∂ 1 ( ψ n ( x )) Examples M ′ n = nM n − 1 Monomial polynomials 1 1 n + 1 T ′ n − 1 T ′ Classical orthogonal n +1 ( x ) − n − 1 ( x ) = 2 T n ( x ) polynomials 2J ′ n ( x ) = J n − 1 ( x ) − J n +1 ( x ) Bessel functions Legendre functions Parabolic cylinder functions 5 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Our framework Families of functions ψ n ( x ) with two special properties Mult by x ( P x ) R ec x 2 ( x ψ n ( x )) = R ec x 1 ( ψ n ( x )) Differentiation ( P ∂ ) R ec ∂ 2 ( ψ ′ n ( x )) = R ec ∂ 1 ( ψ n ( x )) This is our data-structure for ψ n ( x ) 5 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Main Idea Main Idea If ψ n ( x ) satisfies ( P x ) and ( P ∂ ), for any f ( x ) = � a n ψ n ( x ) solution of a linear differential equation with polynomial coefficients, the coefficients a n are cancelled by a linear recurrence relation with polynomial coefficients. 6 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Main Idea Main Idea If ψ n ( x ) satisfies ( P x ) and ( P ∂ ), for any f ( x ) = � a n ψ n ( x ) solution of a linear differential equation with polynomial coefficients, the coefficients a n are cancelled by a linear recurrence relation with polynomial coefficients. Applications: Efficient numerical computation of the coefficients. Computation of closed-form for the coefficients (when it’s possible). 6 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Previous work Clenshaw (1957): numerical scheme to compute the coefficients when ψ n ( x ) = T n ( x ) (Chebyshev series). Lewanowicz (1976-2004): algorithms to compute a recurrence relation when ψ n is an orthogonal or semi-orthogonal polynomial family. Rebillard and Zakrajˇ sek (2006): General algorithm computing a recurrence relation when ψ n is a family of hypergeometric polynomials Benoit and Salvy (2009) : Simple unified presentation and complexity analysis of the previous algorithms using Fractions of recurrence relations when ψ n = T n . New and fast algorithm to compute the Chebyshev recurrence. 7 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion New Results (2011) Simple unified presentation of the previous algorithms using Pairs of recurrence relations. New general algorithm computing the recurrence relation of the coefficients for a Generalized Fourier Series when ψ n ( x ) satisfies ( P x ) and ( P ∂ ). 8 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion II Pairs of Recurrence Relations 9 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Examples: Chebyshev case ( f ( x ) = � u n T n ( x )) Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Basic rules: � a n = u n − 1 + u n +1 xf = ( P x ) a n T n 2 − − → � f ′ = ( P ∂ ) b n − 1 − b n +1 = 2 nu n . b n T n − − → 10 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
Examples: Chebyshev case ( f ( x ) = � u n T n ( x )) Introduction Pairs of Recurrence Relations Recurrences of Smaller Order Conclusion Basic rules: � a n = u n − 1 + u n +1 xf = ( P x ) a n T n 2 − − → � f ′ = ( P ∂ ) b n − 1 − b n +1 = 2 nu n . b n T n − − → Combine: � f ′ + 2 xf = ( P ∂ + 2 P x ) c n − 1 − c n +1 = Rec 1 ( u n ) . c n T n − − − − − − − → Application: Chebyshev series for exp( − x 2 ). 10 / 21 Alexandre Benoit Generalized Fourier Series for Solutions of Linear Differential Equations.
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