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Summer School on Applied Analysis 2011 TU Chemnitz, September 26-30, 2011 Polynomial approximation via de la Vall ee Poussin means Lecture 2: Discrete operators Woula Themistoclakis CNR - National Research Council of Italy Institute for

  1. ◮ COMPARISON WITH LAGRANGE INTERPOLATION Let w = v α,β ∈ L 1 [ − 1 , 1] and u = v γ,δ ∈ C [ − 1 , 1] be such that: α 2 + 1 α 2 + 5 ≤ γ ≤ 4 , 4 β 2 + 1 β 2 + 5 ≤ δ ≤ 4 . 4 Then for all sufficiently large pair of integers n and m = θn ( 0 < θ < 1 fixed) and for each f ∈ C 0 u , we have � [ f − L n ( w, f )] u � ∞ ≤ C log n E n ( f ) u, ∞ Lagrange error: � [ f − ˜ V m n ( w, f )] u � ∞ ≤ CE n − m ( f ) u, ∞ De la V.P. error: where C > 0 is independent of n, f in both the cases. Woula Themistoclakis, Chemnitz, September 26-30, 2011 15

  2. ◮ APPROXIMATION IN L p u Theorem 3: Let 1 ≤ p < ∞ and assume that w = v α,β ∈ L 1 [ − 1 , 1] and u = v γ,δ ∈ L p [ − 1 , 1] , with w u ∈ L p ′ [ − 1 , 1] , satisfy the bounds α 2 + 1 < γ + 1 α 2 + 5  4 − ν p ≤ 4 − ν   0 ≤ ν ≤ 1  for some β 2 + 1 < δ + 1 β 2 + 5 2 4 − ν p ≤ 4 − ν    Then for all n, m ∈ N with m = θn ( 0 < θ < 1 fixed) and each f ∈ L p u (everywhere defined on ] − 1 , 1[ ), we have � n � 1 p � ˜ � λ n ( u p , x n,k ) | f ( x n,k ) | p V n,m ( w, f ) u � p ≤ C k =1 � 1 − x 2 n,k where we recall that λ n ( u p , x n,k ) ∼ u p ( x n,k ) . n Lagrange case L n ( w, f ) : The same estimate holds with p � = 1 and ν = 0 . Woula Themistoclakis, Chemnitz, September 26-30, 2011 16

  3. Error estimates in Sobolev–type spaces u : f ( r − 1) ∈ AC loc , and f ( r ) ϕ r ∈ L p  W p � f ∈ L p � r ( u ) := u  √ � fu � p + � f ( r ) ϕ r u � p , 1 − x 2 � f � W p := ϕ ( x ) :=  r ( u ) r ( u ) , we have E n ( f ) u,p ≤ C n r � f ( r ) ϕ r u � p Note: For any f ∈ W p Theorem 4: Under the assumptions of Theorem 3, for all f ∈ W p r ( u ) , we have C � [ f − ˜ n r � f ( r ) ϕ r u � p , V n,m ( w, f )] u � p ≤ C � f − ˜ V n,m ( w, f ) � W p ≤ n r − s � f � W p r ( u ) , 0 < s ≤ r s ( u ) where C > 0 is independent of f, n, m and 1 ≤ p ≤ ∞ (setting L ∞ u := C 0 u ). Lagrange case: L n ( w, f ) verifies the same estimates, but for p / ∈ { 1 , ∞} and ν = 0 . Woula Themistoclakis, Chemnitz, September 26-30, 2011 17

  4. COMPARISON WITH LAGRANGE INTERPOLATION n � L n ( w, f, x ) := λ n,k K n − 1 ( w, x, x n,k ) f ( x n,k ) , k =1 n ˜ � V n,m ( w, f, x ) := λ n,k H n,m ( w, x, x n,k ) f ( x n,k ) , n > m k =1 ◮ Invariance: ˜ V n,m ( w ) : f → ˜ V n,m ( w, f ) ∈ P n + m − 1 is a quasi–projection ◮ Approximation: ˜ V n,m ( w, f ) solves the “critical” cases p = 1 , ∞ . ◮ Interpolation: Is it true that ˜ V n,m ( w, f, x n,k ) = f ( x n,k ) , k = 1 , .., n ?? Woula Themistoclakis, Chemnitz, September 26-30, 2011 18

  5. Theorem 5 Let w be such that, for all x ∈ ] − 1 , 1[ , we have (3) p n + s ( w, x )+ p n − s ( w, x ) = p n ( w, x ) Q ( x ) , deg ( Q ) ≤ s < n, Then ˜ V n,m ( w, f, x n,i ) = f ( x n,i ) , i = 1 , .., n, holds for all n ≥ m > 0 . Examples: Bernstein–Szego weights defined by | α | = | β | = 1 (1 − x ) α (1 + x ) β , w ( x ) := 2 , Chebyshev weights � 1 1 1 1 − x w ( x ) := √ 1 − x 2 , w ( x ) := 1 + x, deg( p ) ≤ 1 p ( x ) p ( x ) 1 � 1 − x 2 , w ( x ) := deg( p ) ≤ 2 p ( x ) provide polynomials satisfying (3). Woula Themistoclakis, Chemnitz, September 26-30, 2011 19

  6. Proof. Note that we can write m − 1 H n,m ( w, x n,i , x n,j ) = 1 � � � K n + r ( w, x n,i , x n,j ) + K n − ( r +1) ( w, x n,i , x n,j ) 2 m r =0 where: r � K n + r ( w, x n,i , x n,j ) = K n ( w, x n,i , x n,j ) + p n + s ( w, x n,i ) p n + s ( w, x n,j ) , s =1 + = + + r � K n − ( r +1) ( w, x n,i , x n,j ) = K n ( w, x n,i , x n,j ) − p n − s ( w, x n,i ) p n − s ( w, x n,j ) . s =1 Hence by (3) we get p n + s ( w, x n,i ) = − p n − s ( w, x n,i ) , i = 1 , .., n , and the kernel in n ˜ � V n,m ( w, f, x n,i ) = λ n,j H n,m ( w, x n,i , x n,j ) f ( x n,j ) j =1 reduces to H n,m ( w, x n,i , x n,j ) = K n ( w, x n,i , x n,j ) = δ i,j [ λ n,j ] − 1 . ✷ Woula Themistoclakis, Chemnitz, September 26-30, 2011 20

  7. COMPARISON WITH LAGRANGE INTERPOLATION n � L n ( w, f, x ) := λ n,k K n − 1 ( w, x, x n,k ) f ( x n,k ) , k =1 n ˜ � V n,m ( w, f, x ) := λ n,k H n,m ( w, x, x n,k ) f ( x n,k ) , n > m k =1 � L n ( w ) : f → L n ( w, f ) ∈ P n − 1 projection ◮ Invariance: ˜ : f → ˜ V n,m ( w ) V n,m ( w, f ) ∈ P n + m − 1 quasi–projection � � [ f − L n ( w, f )] u � ∞ ≤ C log n E n ( f ) u, ∞ ◮ Approximation: � [ f − ˜ V n,m ( w, f )] u � ∞ ≤ CE n − m ( f ) u, ∞ � for all w = v α,β L n ( w, f, x n,k ) = f ( x n,k ) , ◮ Interpolation: ˜ | α | = | β | = 1 V n,m ( w, f, x n,k ) = f ( x n,k ) , 2 , n ≥ m Woula Themistoclakis, Chemnitz, September 26-30, 2011 21

  8. De la Vall´ ee Poussin type polynomial spaces DEF : S n,m ( w ) := span { λ n,k H n,m ( w, x, x n,k ) : k = 1 , . . . , n } ◮ Interpolation property: λ n,k H n,m ( w, x n,h , x n,k ) = δ h,k ⇓ dim S n,m ( w ) = n ◮ Invariance property: ˜ V n,m ( w, P ) = P, deg( P ) ≤ n − m ⇓ P n − m ⊂ S n,m ( w ) ⊂ P n + m − 1 Theorem 6 : In the interpolating case, w = v α,β with | α | = | β | = 1 2 , the operator ˜ V n,m ( w ) : f → ˜ V n,m ( w, f ) is a projection on S n,m ( w ) , i.e. we have f ∈ S n,m ( w ) ⇔ f = ˜ V n,m ( w, f ) Woula Themistoclakis, Chemnitz, September 26-30, 2011 22

  9. INTERPOLATING BASIS OF S n,m ( w ) : � � Φ m S n,m ( w ) := span n,k ( w, x ) := λ n,k H n,m ( w, x, x n,k ) , k = 1 , . . . , n n ˜ � f ( x n,k )Φ m V n,m ( w, f, x ) = n,k ( w, x ) De la V. P. interpolating polynomial: k =1 Under the assumptions of Theorem 3, for all a k ∈ R , k = 1 , .., n , we have � n � 1  p � n  � � ��  λ k ( u p , x n,k ) | a k | p if 1 ≤ p < ∞  � �  � a k Φ m � u n,k ( w ) ∼ � � k =1 � � �  k =1 p  1 ≤ k ≤ n | a k | u ( x n,k ) max if p = ∞   i.e. { Φ m n,k ( w ) } k is a Marcinkiewicz basis in L p u , for all 1 ≤ p ≤ ∞ . Woula Themistoclakis, Chemnitz, September 26-30, 2011 23

  10. ORTHOGONAL BASIS OF S n,m ( w )  p k ( w ) if 0 ≤ k ≤ n − m  q k ( w ) := m + n − k p k ( w ) − m − n + k p 2 n − k ( w ) if n − m < k < n  2 m 2 m Theorem 7: The set { q k ( w ) } k is an orthogonal basis of S n,m ( w ) , i.e. we have S n,m ( w ) := span { q k ( w ) : k = 0 , 1 , . . . , n − 1 } with  1 if 0 ≤ k ≤ n − m � 1  q h ( w, x ) q k ( w, x ) w ( x ) dx = δ h,k · m 2 +( n − k ) 2 if n − m < k < n − 1  2 m 2 De la Vall´ ee Poussin interpolating polynomial: � n n − 1 � ˜ � � V n,m ( w, f, x ) = λ n,i p k ( w, x n,i ) f ( x n,i ) q k ( w, x ) i =1 k =0 Woula Themistoclakis, Chemnitz, September 26-30, 2011 24

  11. Proof. We are going to state the basis transformation n − 1 � Φ m n,k ( w, x ) = λ n,k p j ( w, x n,k ) q j ( w, x ) , k = 1 , . . . , n. j =0 Recall that n + m − 1 � Φ m µ m n,k ( w, x ) := λ n,k H n,m ( w, x, x n,k ) = λ n,k s n,j p j ( w, x n,k ) p j ( w, x ) j =0  n − m n − 1 n + m − j � � = λ n,k p j ( w, x n,k ) p j ( w, x ) + p j ( w, x n,k ) p j ( w, x )  2 m j =0 j = n − m +1  n + m − 1 n + m − j � + p j ( w, x n,k ) p j ( w, x )  2 m j = n +1 i.e., by changing the summation variables, we have Woula Themistoclakis, Chemnitz, September 26-30, 2011 25

  12.  n − m m − 1 m + s � � Φ m = λ n,k p j ( w, x n,k ) p j ( w ) + 2 m p n − s ( w, x n,k ) p n − s ( w ) n,k  s =1 j =0 m − 1 � m − s � + 2 m p n + s ( w, x n,k ) p n + s ( w ) s =1 and using p n + s ( w, x n,k ) = − p n − s ( w, x n,k ) , we get n − m � Φ m s = λ n,k p j ( w, x n,k ) p j ( w ) + n,k j =0 m − 1 � m + s 2 m p n − s ( w ) − m − s � � + λ n,k p n − s ( w, x n,k ) 2 m p n + s ( w ) s =1 n − 1 � = λ n,k p j ( w, x n,k ) q j ( w ) . ✷ j =0 Woula Themistoclakis, Chemnitz, September 26-30, 2011 26

  13. COMPARISON WITH LAGRANGE INTERPOLATION Chebyshev case: n − 1 � L n ( w, f, x ) := ˜ c n,k ( w, f ) p k ( w, x ) , k =0 n − 1 ˜ � V n,m ( w, f, x ) := ˜ c n,k ( w, f ) q k ( w, x ) , n > m k =0 � L n ( w, f, x n,k ) = f ( x n,k ) , k = 1 , . . . , n ◮ Interpolation: ˜ V n,m ( w, f, x n,k ) = f ( x n,k ) , k = 1 , . . . , n � L n ( w ) : f → L n ( w, f ) ∈ P n − 1 projection ◮ Invariance: ˜ : f → ˜ V n,m ( w ) V n,m ( w, f ) ∈ S n,m ( w ) projection � � [ f − L n ( w, f )] u � ∞ ≤ C log n E n ( f ) u, ∞ ◮ Approximation: � [ f − ˜ V n,m ( w, f )] u � ∞ ≤ CE n − m ( f ) u, ∞ Woula Themistoclakis, Chemnitz, September 26-30, 2011 27

  14. Lagrange interpolation for n = 50 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 28

  15. θ n, with θ = 0.1 De la V.P. interpolation for n = 50 and m = 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 29

  16. θ n, with θ = 0.2 De la V.P. interpolation for n = 50 and m = 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 30

  17. θ n, with θ = 0.3 De la V.P. interpolation for n = 50 and m = 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 31

  18. θ n, with θ = 0.4 De la V.P. interpolation for n = 50 and m = 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 32

  19. θ n, with θ = 0.5 De la V.P. interpolation for n = 50 and m = 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 33

  20. θ n, with θ = 0.6 De la V.P. interpolation for n = 50 and m = 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 34

  21. θ n, with θ = 0.7 De la V.P. interpolation for n = 50 and m = 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 35

  22. θ n, with θ = 0.8 De la V.P. interpolation for n = 50 and m = 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 36

  23. θ n, with θ = 0.9 De la V.P. interpolation for n = 50 and m = 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 37

  24. θ n, with θ = 1 De la V.P. interpolation for n = 50 and m = 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 38

  25. θ = 0.1 Error curves by Lagrange (in red) and de la V.P. interpolation for 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 39

  26. θ = 0.2 Error curves by Lagrange (in red) and de la V.P. interpolation for 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 40

  27. θ = 0.3 Error curves by Lagrange (in red) and de la V.P. interpolation for 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 41

  28. θ = 0.4 Error curves by Lagrange (in red) and de la V.P. interpolation for 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 42

  29. θ = 0.5 Error curves by Lagrange (in red) and de la V.P. interpolation for 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 43

  30. θ = 0.6 Error curves by Lagrange (in red) and de la V.P. interpolation for 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 44

  31. θ = 0.7 Error curves by Lagrange (in red) and de la V.P. interpolation for 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 45

  32. θ = 0.8 Error curves by Lagrange (in red) and de la V.P. interpolation for 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 46

  33. θ = 0.9 Error curves by Lagrange (in red) and de la V.P. interpolation for 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 47

  34. θ = 1 Error curves by Lagrange (in red) and de la V.P. interpolation for 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 48

  35. Lagrange interpolation for n = 20 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 49

  36. θ n, with θ = 0.1 De la V.P. interpolation for n = 20 and m = 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 50

  37. θ n, with θ = 0.2 De la V.P. interpolation for n = 20 and m = 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 51

  38. θ n, with θ = 0.3 De la V.P. interpolation for n = 20 and m = 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 52

  39. θ n, with θ = 0.4 De la V.P. interpolation for n = 20 and m = 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 53

  40. θ n, with θ = 0.5 De la V.P. interpolation for n = 20 and m = 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 54

  41. θ n, with θ = 0.6 De la V.P. interpolation for n = 20 and m = 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 55

  42. θ n, with θ = 0.7 De la V.P. interpolation for n = 20 and m = 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 56

  43. θ n, with θ = 0.8 De la V.P. interpolation for n = 20 and m = 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 57

  44. θ n, with θ = 0.9 De la V.P. interpolation for n = 20 and m = 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 58

  45. θ n, with θ = 1 De la V.P. interpolation for n = 20 and m = 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 59

  46. θ = 0.1 Error curves by Lagrange (in red) and de la V.P. interpolation for 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 60

  47. θ = 0.2 Error curves by Lagrange (in red) and de la V.P. interpolation for 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 61

  48. θ = 0.3 Error curves by Lagrange (in red) and de la V.P. interpolation for 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 62

  49. θ = 0.4 Error curves by Lagrange (in red) and de la V.P. interpolation for 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 63

  50. θ = 0.5 Error curves by Lagrange (in red) and de la V.P. interpolation for 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 64

  51. θ = 0.6 Error curves by Lagrange (in red) and de la V.P. interpolation for 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 65

  52. θ = 0.7 Error curves by Lagrange (in red) and de la V.P. interpolation for 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 66

  53. θ = 0.8 Error curves by Lagrange (in red) and de la V.P. interpolation for 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 67

  54. θ = 0.9 Error curves by Lagrange (in red) and de la V.P. interpolation for 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 68

  55. θ = 1 Error curves by Lagrange (in red) and de la V.P. interpolation for 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 69

  56. Fundamental Lagrange polynomial for n = 50 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 70

  57. De la VP kernel polynomials for n = 50 and m = θ n, with θ = 0.1 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 71

  58. De la VP kernel polynomials for n = 50 and m = θ n, with θ = 0.2 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 72

  59. De la VP kernel polynomials for n = 50 and m = θ n, with θ = 0.3 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 73

  60. De la VP kernel polynomials for n = 50 and m = θ n, with θ = 0.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 74

  61. De la VP kernel polynomials for n = 50 and m = θ n, with θ = 0.5 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 75

  62. De la VP kernel polynomials for n = 50 and m = θ n, with θ = 0.6 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 76

  63. De la VP kernel polynomials for n = 50 and m = θ n, with θ = 0.7 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 77

  64. De la VP kernel polynomials for n = 50 and m = θ n, with θ = 0.8 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 78

  65. De la VP kernel polynomials for n = 50 and m = θ n, with θ = 0.9 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 79

  66. Fejer kernel polynomials for n = 50 and m = θ n, with θ = 1 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 80

  67. Lagrange interpolation for n = 50 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 81

  68. θ n, with θ = 0.1 De la V.P. interpolation for n = 50 and m = 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 82

  69. θ n, with θ = 0.2 De la V.P. interpolation for n = 50 and m = 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 83

  70. θ n, with θ = 0.3 De la V.P. interpolation for n = 50 and m = 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 84

  71. θ n, with θ = 0.4 De la V.P. interpolation for n = 50 and m = 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 85

  72. θ n, with θ = 0.5 De la V.P. interpolation for n = 50 and m = 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 86

  73. θ n, with θ = 0.6 De la V.P. interpolation for n = 50 and m = 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 87

  74. θ n, with θ = 0.7 De la V.P. interpolation for n = 50 and m = 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 88

  75. θ n, with θ = 0.8 De la V.P. interpolation for n = 50 and m = 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 89

  76. θ n, with θ = 0.9 De la V.P. interpolation for n = 50 and m = 2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 90

  77. θ = 0.1 Error curves by Lagrange (in red) and de la V.P. interpolation for 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 91

  78. θ = 0.2 Error curves by Lagrange (in red) and de la V.P. interpolation for 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 92

  79. θ = 0.3 Error curves by Lagrange (in red) and de la V.P. interpolation for 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 93

  80. θ = 0.4 Error curves by Lagrange (in red) and de la V.P. interpolation for 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 94

  81. θ = 0.5 Error curves by Lagrange (in red) and de la V.P. interpolation for 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 95

  82. θ = 0.6 Error curves by Lagrange (in red) and de la V.P. interpolation for 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 96

  83. θ = 0.7 Error curves by Lagrange (in red) and de la V.P. interpolation for 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 97

  84. θ = 0.8 Error curves by Lagrange (in red) and de la V.P. interpolation for 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 98

  85. θ = 0.9 Error curves by Lagrange (in red) and de la V.P. interpolation for 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 99

  86. Lagrange interpolation for n = 100 3.5 3 2.5 2 1.5 1 0.5 0 −0.5 −1 −0.5 0 0.5 1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 100

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