Multivariate Christoffel functions and hyperinterpolation 1 Stefano - PowerPoint PPT Presentation

Multivariate Christoffel functions and hyperinterpolation 1 Stefano De Marchi Department of Mathematics - University of Padova Goettingen - December 2, 2014 1 Joint work with Alvise Sommariva and Marco Vianello

Motivation Len Bos, Multivariate interpolation and polynomial inequalities, Ph.D. course held in 2001 at the University of Padova (unpublished notes) 2 of 26

Motivation Len Bos, Multivariate interpolation and polynomial inequalities, Ph.D. course held in 2001 at the University of Padova (unpublished notes) He proved by means of the bivariate Christoffel-Darboux formula of Xu that the Lebesgue constant of the Morrow-Patterson points, Λ MP = O ( n 6 ) . n 2 of 26

Motivation Len Bos, Multivariate interpolation and polynomial inequalities, Ph.D. course held in 2001 at the University of Padova (unpublished notes) He proved by means of the bivariate Christoffel-Darboux formula of Xu that the Lebesgue constant of the Morrow-Patterson points, Λ MP = O ( n 6 ) . n Morrow-Patterson points were the basis of inspiration of the Padua Points. 2 of 26

Outline 1 The problem Estimates for Christofell functions 2 Disk and ball Square and cube 3 Upper bounds for Lebesgue constants The d -dimensional ball The d -dimensional cube The Morrow-Patterson points 4 References 3 of 26

Introduction Notation n ( K )) := ( n + d K ⊂ R d , P d n ( K ) , N = dim ( P d d ); 1 4 of 26

Introduction Notation n ( K )) := ( n + d K ⊂ R d , P d n ( K ) , N = dim ( P d d ); 1 K n ( x , y ) : reproducing kernel of P d n ( K ) in L 2 d µ ( K ) ( µ a positive 2 measure on K ) with representation (cf. Dunkl and Xu 2001, § 3.5) N � p j ( x ) p j ( y ) , x , y ∈ R d , K n ( x , y ) = (1) j = 1 where { p j } is any orthonormal basis of P d n ( K ) in L 2 d µ ( K ) . The function N � p 2 K n ( x , x ) = j ( x ) (2) j = 1 is known as the (reciprocal of) the n -th Christoffel function of µ on K . 4 of 26

Introduction Hyperinterpolation operator Definition Hyperinterpolation of multivariate continuous functions, on compact subsets or manifolds, is a discretized orthogonal projection on polynomial subspaces [Sloan JAT1995]. It requires 3 main ingredients 5 of 26

Introduction Hyperinterpolation operator Definition Hyperinterpolation of multivariate continuous functions, on compact subsets or manifolds, is a discretized orthogonal projection on polynomial subspaces [Sloan JAT1995]. It requires 3 main ingredients a good cubature formula (positive weights and high precision); 1 a good formula for representing the reproducing kernel (accurate 2 and efficient); a slow increase of the Lebegsue constant (which is the operator 3 norm). 5 of 26

Introduction Hyperinterpolation operator Definition Hyperinterpolation of multivariate continuous functions, on compact subsets or manifolds, is a discretized orthogonal projection on polynomial subspaces [Sloan JAT1995]. It requires 3 main ingredients a good cubature formula (positive weights and high precision); 1 a good formula for representing the reproducing kernel (accurate 2 and efficient); a slow increase of the Lebegsue constant (which is the operator 3 norm). Practically It is a total-degree polynomial approximation of multivariate continuous functions, given by a truncated Fourier expansion in o.p. for the given domain 5 of 26

Introduction Initial observation We observe the following fact. Let { a n } ∈ R + be a sequence s.t. � a n ≥ C n ( d µ, K ) = max x ∈ K K n ( x , x ) (3) Let � � � � P d L n : C ( K ) , � · � L ∞ ( K ) → n , � · � L 2 (4) d µ ( K ) uniformly bounded operators, i.e. ∃ M > 0 s.t. for every n �L n f � L 2 d µ ( K ) �L n � = sup ≤ M . � f � L ∞ ( K ) f � 0 Then this estimate holds: �L n f � L ∞ ( K ) �L n � ∞ = sup ≤ a n M . (5) � f � L ∞ ( K ) f � 0 6 of 26

Lebesgue constant of the hyperinterpolation operator Given cubature formula ( X , w ) for µ , exact in P d 2 n ( K ) , with nodes X = X n = { ξ i ( n ) , i = 1 , . . . , V} ⊂ K and positive weights w = w n = { w i ( n ) , i = 1 , . . . , V} , V ≥ N = dim ( P d n ( K )) , { p j , j = 1 , . . . , N } be any orthonormal basis of P d n ( K ) in L 2 d µ ( K ) . hyperinterpolation operator is the discretized orthogonal projection L n : C ( K ) → P d n ( K ) defined as N � L n f ( x ) = � f , p j � ℓ 2 w ( X ) p j ( x ) , j = 1 where ℓ 2 w ( X ) is equipped with the scalar product V � � f , g � = w i f ( ξ i ) g ( ξ i ) . i = 1 7 of 26

”Lebesgue constant” of the hyperinterpolation operator Corollary 1 Assume that (3) holds, then � �L n � ∞ ≤ a n µ ( K ) . (6) Proof. Following Sloan [JAT95], we can write by exactness in P d 2 n ( K ) and the Pythagorean theorem in ℓ 2 w ( X ) � � V � � w i f 2 ( ξ i ) �L n f � L 2 d µ ( K ) = �L n f � ℓ 2 w ( X ) ≤ � f � ℓ 2 w ( X ) = i = 1 � � V � � � � ≤ w i � f � ℓ ∞ ( X ) = µ ( K ) � f � ℓ ∞ ( X ) ≤ µ ( K ) � f � L ∞ ( K ) , i = 1 � so that in Proposition 1 we can take M = µ ( K ) . � 8 of 26

Estimates disk and ball, K = B d Here we use the Gegenbauer measure W λ ( x ) = ( 1 − | x | 2 ) λ − 1 / 2 , λ > − 1 2 , (7) 9 of 26

Estimates disk and ball, K = B d Here we use the Gegenbauer measure W λ ( x ) = ( 1 − | x | 2 ) λ − 1 / 2 , λ > − 1 2 , (7) Bos in [NZJM,94] proved � �� n + d � � n + d − 1 �� 2 = O ( n d / 2 ) , C n ( W 0 ( x ) d x , B d ) ≤ + (8) ω d d d ω d being the surface area of the unit sphere S d ⊂ R d + 1 . Later [Bloom, Bos, Levenberg, APM12] showed that C n has polynomial growth on the ball for d µ = W λ ( x ) d x , λ ≥ 0. No explicit bounds were provided! 9 of 26

Estimates formulas for K = B 2 Main ingredient: Zernike polynomials (see [Carnicer, God´ es NA14]), orthogonal basis on the disk w.r.t. Lebesgue measure (used in optics)  � 2 ( h + 1 ) R m h ( r ) cos ( m θ ) , m ≥ 0    α m   Z m ˆ  h ( r , θ ) = (9)    �  2 ( h + 1 )  R m  h ( r ) sin ( m θ ) , m < 0   α m for 0 ≤ h ≤ n , | m | ≤ h , h − m ∈ 2 Z , where  2 , m = 0    α m = (10)    1 , m � 0   R m h ( r ) = ( − 1 ) ( h − m ) / 2 r m P m , 0 ( h − m ) / 2 ( 1 − 2 r 2 ) (11) and P m , 0 is the corresponding Jacobi polynomial of degree j . j 10 of 26

Estimates formulas for the disk, K = B 2 Relevant property: for 0 ≤ h ≤ n , | m | ≤ h , h − m ∈ 2 Z � 2 h + 2 | ˆ Z m h ( r , θ ) | ≤ , x = ( r cos ( θ ) , r sin ( θ )) ∈ B 2 . π n n h ( r , θ )) 2 ≤ 1 � � � � (ˆ Z m K n ( x , x ) = ( 2 h + 2 ) π h = 0 | m |≤ h , h − m ∈ 2 Z h = 0 | m |≤ h , h − m ∈ 2 Z n = 1 ( 2 h + 2 )( n − h + 1 ) = 1 � 3 π ( n + 1 )( n + 2 )( n + 3 ) , π h = 0 and hence 1 � ( n + 1 )( n + 2 )( n + 3 ) = O ( n 3 / 2 ) . C n ( d x , B 2 ) ≤ √ (12) 3 π 11 of 26

Estimates formulas for the cube, K = [ − 1 , 1 ] d Jacobi measure d � ( 1 − x i ) α ( 1 + x i ) β , α, β > − 1 , d µ = W α,β ( x ) d x , W α,β ( x ) = i = 1 (13) Total-degree orthonormal product basis d � Π α,β P α,β ˆ k ( x ) = k i ( x i ) , 0 ≤ | k | ≤ n , (14) i = 1 i = 1 k i , and ˆ P α,β where k = ( k 1 , . . . , k d ) with k i ≥ 0 and | k | = � d m denotes the m -th degree polynomial of the univariate orthonormal Jacobi basis with parameters α and β . 12 of 26

Estimates formulas for K = [ − 1 , 1 ] d For max { α, β } ≥ − 1 / 2, max | ˆ P α,β k i | at ± 1, then � | ˆ P α,β m ( t ) | ≤ | ˆ P α,β ( 2 m + α + β + 1 )Γ( m + α + β + 1 )Γ( m + q + 1 ) m ( sign ( α − β )) | = 2 α + β + 1 m ! Γ( m + min { α, β } + 1 ) ≤ c ( α, β ) m q + 1 / 2 , t ∈ [ − 1 , 1 ] , q = max { α, β } ≥ − 1 (15) 2 , 13 of 26

Estimates formulas for K = [ − 1 , 1 ] d � 2 � � Π α,β x ∈ [ − 1 , 1 ] d K n ( x , x ) = max max k ( x ) x ∈ [ − 1 , 1 ] d 0 ≤| k |≤ n d d � 2 ≤ ( c ( α, β )) 2 d � � � � k 2 q + 1 � ˆ P α,β = k i ( sign ( α − β )) i 0 ≤| k |≤ n i = 1 0 ≤| k |≤ n i = 1 n − � d − 1 j = 1 k j n n − k 1 � � � k 2 q + 1 k 2 q + 1 k 2 q + 1 = ( c ( α, β )) 2 d = O ( n ( 2 q + 2 ) d ) , · · · 1 2 d k 1 = 0 k 2 = 0 k d = 0 which gives the qualitative bound C n ( W α,β ( x ) d x , [ − 1 , 1 ] d ) = O ( n ( q + 1 ) d ) . (16) 14 of 26

Special cases α = β = 0, Legendre polynomials � 2 m + 1 | ˆ P 0 , 0 m ( t ) | ≤ ˆ P 0 , 0 m ( 1 ) = , t ∈ [ − 1 , 1 ] , 2 from which we have n − � d − 1 j = 1 kj n − k 1 d n � 2 1 � � � P 0 , 0 � � � ˆ x ∈ [ − 1 , 1 ] d K n ( x , x ) = max ki ( 1 ) = ( 2 k 1 + 1 ) ( 2 k 2 + 1 ) · · · ( 2 k d + 1 ) , (17) 2 d i = 1 k 1 = 0 k 2 = 0 kd = 0 0 ≤| k |≤ n 15 of 26