Polynomial splines as examples of Chebyshevian splines - PowerPoint PPT Presentation

Polynomial splines as examples of Chebyshevian splines Marie-Laurence Mazure Laboratoire Jean Kuntzmann, Université Joseph Fourier, Grenoble SC2011 – S. Margherita di Pula, October 10-14, 2011 1 / 24

Outline 2 / 24

Outline Extended Chebyshev spaces 2 / 24

Outline Extended Chebyshev spaces Chebyshevian splines 2 / 24

Outline Extended Chebyshev spaces Chebyshevian splines Geometrically continuous polynomial splines 2 / 24

Outline Extended Chebyshev spaces ( Th . A ) Chebyshevian splines Geometrically continuous polynomial splines 2 / 24

Outline Extended Chebyshev spaces ( Th . A ) Chebyshevian splines ( Th . B ) Geometrically continuous polynomial splines 2 / 24

Outline Extended Chebyshev spaces ( Th . A ) Chebyshevian splines ( Th . B ) Geometrically continuous polynomial splines ( ⇐ A + B ) 2 / 24

Extended Chebyshev spaces (EC-spaces) 3 / 24

Extended Chebyshev spaces (EC-spaces) natural generalisation of polynomial spaces 3 / 24

Extended Chebyshev spaces (EC-spaces) natural generalisation of polynomial spaces same properties as polynomial spaces (up to 1 ? exception) but DIFFICULT TO PROVE 3 / 24

Extended Chebyshev spaces (EC-spaces) natural generalisation of polynomial spaces same properties as polynomial spaces (up to 1 ? exception) but DIFFICULT TO PROVE advantages : 3 / 24

Extended Chebyshev spaces (EC-spaces) natural generalisation of polynomial spaces same properties as polynomial spaces (up to 1 ? exception) but DIFFICULT TO PROVE advantages : offer more possibilities in Approximation or Geometric Design 3 / 24

Extended Chebyshev spaces (EC-spaces) natural generalisation of polynomial spaces same properties as polynomial spaces (up to 1 ? exception) but DIFFICULT TO PROVE advantages : offer more possibilities in Approximation or Geometric Design enable us to better understand polynomial spaces 3 / 24

Extended Chebyshev spaces (EC-spaces) natural generalisation of polynomial spaces same properties as polynomial spaces (up to 1 ? exception) but DIFFICULT TO PROVE advantages : offer more possibilities in Approximation or Geometric Design enable us to better understand polynomial spaces or even to obtain new results concerning them e.g., HERE 3 / 24

EC-spaces 4 / 24

EC-spaces I interval, E ⊂ C n ( I ) ( n + 1 ) -dimensional 4 / 24

EC-spaces I interval, E ⊂ C n ( I ) ( n + 1 ) -dimensional Definition : E is an EC-space on I is any non-zero F ∈ E vanishes at most n times in I , counting multiplicities 4 / 24

EC-spaces I interval, E ⊂ C n ( I ) ( n + 1 ) -dimensional Definition : E is an EC-space on I is any non-zero F ∈ E vanishes at most n times in I , counting multiplicities Examples : 1 1 , x , . . . , x n − 2 , cosh x , sinh x span an EC-space on I = R ; 2 1 , x , . . . , x n − 2 , cos x , sin x span an EC-space on any I = [ a , a + 2 π [ (for n ≥ 2). 3 . . . 4 / 24

EC-spaces I interval, E ⊂ C n ( I ) ( n + 1 ) -dimensional Definition : E is an EC-space on I is any non-zero F ∈ E vanishes at most n times in I , counting multiplicities Examples : 1 1 , x , . . . , x n − 2 , cosh x , sinh x span an EC-space on I = R ; 2 1 , x , . . . , x n − 2 , cos x , sin x span an EC-space on any I = [ a , a + 2 π [ (for n ≥ 2). 3 . . . S. Karlin – L.L. Schumaker – T. Lyche – N. Dyn – A. Ron – G. Mühlbach – H. Pottmann – P.J. Barry – D. Bister – H. Prautzsch – J. Carnicer – J.-M. Peña – P. Costantini – C. Manni –. . . 4 / 24

EC-spaces and weight functions 5 / 24

EC-spaces and weight functions ( w 0 , . . . , w n ) system of weight functions on I : for i = 0 , . . . , n , w i ∈ C n − i ( I ) and is positive on I 5 / 24

EC-spaces and weight functions ( w 0 , . . . , w n ) system of weight functions on I : for i = 0 , . . . , n , w i ∈ C n − i ( I ) and is positive on I associated generalised derivatives : D = ordinary differentiation L 0 F := F L i F := DL i − 1 F , i = 1 , . . . , n , w 0 w i 5 / 24

EC-spaces and weight functions ( w 0 , . . . , w n ) system of weight functions on I : for i = 0 , . . . , n , w i ∈ C n − i ( I ) and is positive on I associated generalised derivatives : D = ordinary differentiation L 0 F := F L i F := DL i − 1 F , i = 1 , . . . , n , w 0 w i E := { F ∈ C n ( I ) | L n F constant on I } is an ( n + 1 ) -dim. EC-space on I , denoted E = EC ( w 0 , . . . , w n ) 5 / 24

EC-spaces and weight functions ( w 0 , . . . , w n ) system of weight functions on I : for i = 0 , . . . , n , w i ∈ C n − i ( I ) and is positive on I associated generalised derivatives : D = ordinary differentiation L 0 F := F L i F := DL i − 1 F , i = 1 , . . . , n , w 0 w i E := { F ∈ C n ( I ) | L n F constant on I } is an ( n + 1 ) -dim. EC-space on I , denoted E = EC ( w 0 , . . . , w n ) Ex : on I = R , for w 0 = w 1 = · · · = w n = 1 I , EC ( w 0 , . . . , w n ) = P n 5 / 24

E = ( n + 1 ) -dimensional EC-space on I = [ a , b ] 6 / 24

E = ( n + 1 ) -dimensional EC-space on I = [ a , b ] Theorem - [H.Pottmann 93, MLM 05] If I is closed and bounded , then E can be written as E = EC ( w 0 , . . . , w n ) 6 / 24

E = ( n + 1 ) -dimensional EC-space on I = [ a , b ] Theorem - [H.Pottmann 93, MLM 05] If I is closed and bounded , then E can be written as E = EC ( w 0 , . . . , w n ) THEOREM A ONE CAN NOW FIND ALL POSSIBLE SYSTEMS OF WEIGHT FUNCTIONS ON [ a , b ] SUCH THAT E = EC ( w 0 , . . . , w n ) 6 / 24

E = ( n + 1 ) -dimensional EC-space on I = [ a , b ] Theorem - [H.Pottmann 93, MLM 05] If I is closed and bounded , then E can be written as E = EC ( w 0 , . . . , w n ) THEOREM A ONE CAN NOW FIND ALL POSSIBLE SYSTEMS OF WEIGHT FUNCTIONS ON [ a , b ] SUCH THAT E = EC ( w 0 , . . . , w n ) MLM, Finding all systems of weight functions associated with a given Extended Chebyshev space, J. Approx. Theory , 2011 6 / 24

Weight functions for E := P 2 restricted to I := [ 0 , 1 ] 7 / 24

Weight functions for E := P 2 restricted to I := [ 0 , 1 ] ( B 0 , B 1 , B 2 ) : Bernstein basis of degree 2 7 / 24

Weight functions for E := P 2 restricted to I := [ 0 , 1 ] ( B 0 , B 1 , B 2 ) : Bernstein basis of degree 2 Select any α 0 , α 1 , α 2 > 0 and take w 0 := � 2 i = 0 α i B i 7 / 24

Weight functions for E := P 2 restricted to I := [ 0 , 1 ] ( B 0 , B 1 , B 2 ) : Bernstein basis of degree 2 Select any α 0 , α 1 , α 2 > 0 and take w 0 := � 2 i = 0 α i B i I = � 2 α i B i → Bernstein basis of L 0 E := { F w 0 | F ∈ E } 1 i = 0 w 0 7 / 24

Weight functions for E := P 2 restricted to I := [ 0 , 1 ] ( B 0 , B 1 , B 2 ) : Bernstein basis of degree 2 Select any α 0 , α 1 , α 2 > 0 and take w 0 := � 2 i = 0 α i B i I = � 2 α i B i → Bernstein basis of L 0 E := { F w 0 | F ∈ E } 1 i = 0 w 0 DL 0 E = EC-space of dimension 2 on I , Bernstein-like basis � α 0 B 0 � � α 1 B 1 � V 0 := − D V 1 := D , w 0 w 0 7 / 24

Weight functions for E := P 2 restricted to I := [ 0 , 1 ] ( B 0 , B 1 , B 2 ) : Bernstein basis of degree 2 Select any α 0 , α 1 , α 2 > 0 and take w 0 := � 2 i = 0 α i B i I = � 2 α i B i → Bernstein basis of L 0 E := { F w 0 | F ∈ E } 1 i = 0 w 0 DL 0 E = EC-space of dimension 2 on I , Bernstein-like basis � α 0 B 0 � � α 1 B 1 � V 0 := − D V 1 := D , w 0 w 0 Select any β 0 , β 1 > 0 and take w 1 := � 1 i = 0 β i V i 7 / 24

Weight functions for E := P 2 restricted to I := [ 0 , 1 ] I = � 1 β i V i w 1 → Bern. basis of L 1 E := { F 1 w 1 | F ∈ DL 0 E } i = 0 8 / 24

Weight functions for E := P 2 restricted to I := [ 0 , 1 ] I = � 1 β i V i w 1 → Bern. basis of L 1 E := { F 1 w 1 | F ∈ DL 0 E } i = 0 DL 1 E := EC-space of dimension 1 on I , with Bernstein-like basis � β 1 V 1 � V 0 := D w 1 8 / 24

Weight functions for E := P 2 restricted to I := [ 0 , 1 ] I = � 1 β i V i w 1 → Bern. basis of L 1 E := { F 1 w 1 | F ∈ DL 0 E } i = 0 DL 1 E := EC-space of dimension 1 on I , with Bernstein-like basis � β 1 V 1 � V 0 := D w 1 Select any γ 0 > 0 and take w 2 := γ 0 V 0 8 / 24

Weight functions for E := P 2 restricted to I := [ 0 , 1 ] I = � 1 β i V i w 1 → Bern. basis of L 1 E := { F 1 w 1 | F ∈ DL 0 E } i = 0 DL 1 E := EC-space of dimension 1 on I , with Bernstein-like basis � β 1 V 1 � V 0 := D w 1 Select any γ 0 > 0 and take w 2 := γ 0 V 0 Theorem - This provides us with all ways to write E as E = EC ( w 0 , w 1 , w 2 ) . 8 / 24

Chebyshevian splines : ingredients 9 / 24

Chebyshevian splines : ingredients 1 a sequence of knots t k < t k + 1 , k ∈ Z ; 9 / 24

Chebyshevian splines : ingredients 1 a sequence of knots t k < t k + 1 , k ∈ Z ; 2 a sequence of section-spaces : for each k ∈ Z , E k ⊂ C n ([ t k , t k + 1 ]) contains constants and D E k is an n -dimensional EC-space on [ t k , t k + 1 ] ; 9 / 24

Chebyshevian splines : ingredients 1 a sequence of knots t k < t k + 1 , k ∈ Z ; 2 a sequence of section-spaces : for each k ∈ Z , E k ⊂ C n ([ t k , t k + 1 ]) contains constants and D E k is an n -dimensional EC-space on [ t k , t k + 1 ] ; 3 a sequence M k , k ∈ Z , of connection matrices : for each k ∈ Z , M k is a lower triangular matrix of order ( n − 1 ) with positive diagonal. 9 / 24

Chebyshevian splines : definition 10 / 24