Polynomial approximation via de la Vallée Poussin means

Transcription

1 Summer School on Applied Analysis 2 TU Chemnitz, September 26-3, 2 Polynomial approximation via de la Vallée Poussin means Lecture 2: Discrete operators Woula Themistoclakis CNR - National Research Council of Italy Institute for Computational Applications Mauro Picone, Naples, Italy. Woula Themistoclakis, Chemnitz, September 26-3, 2

2 De la Vallée Poussin means: V n,m (w, f, x) := 2m where S n (w, f, x) := c k (w, f) := n c k (w, f)p k (w, x) k= p k (w, y)f(y)w(y)dy n+m k=n m S k (w, f, x) Fourier sum Fourier coefficients w(x) := v α,β (x) = ( x) α ( + x) β, α, β >, being a Jacobi weight and {p j (w, x)} j the corresponding system of orthonormal Jacobi polynomials Quasi-projection: V n,m (w, P ) = P, P P n m Near best polynomial: (f V n,m (w, f))u p CE n m (f) u,p holds for any p, m = θn (θ ], [ fixed) and suitable u, w. Woula Themistoclakis, Chemnitz, September 26-3, 2 2

3 Other forms: V n,m (w, f, x) = V n,m (w, f, x) = n+m k= H n,m (w, x, y)f(y)w(y)dy () µ m n,k c k (w, f)p k (w, x) (2) n+m H n,m (w, x, y) := K r (w, x, y) 2m r=n m r K r (w, x, y) := p j (w, x)p j (w, y) j= de la Vallée Poussin kernel Darboux kernel c k (w, f) := µ m n,k := p k (w, y)f(y)w(y)dy if k n m, n + m k 2m if n m < k < n + m. Fourier coefficients Woula Themistoclakis, Chemnitz, September 26-3, 2 3

4 Lagrange interpolation at the zeros of orthogonal polynomials Take the Fourier sum: S n (w, f, x) := K n (w, x, y)f(y)w(y)dy Apply the Gaussian rule: g(x)w(x)dx = n λ n,j g(x n,j ), j= g P 2n In this way we obtain the Lagrange polynomial of degree n interpolating f at the n zeros {x n,j } j of p n (w), i.e. L n (w, f, x) := n λ n,j K n (w, x, x n,j )f(x n,j ) j= with λ n,j K n (w, x n,i, x n,j ) = δ i,j, j =,..., n. Woula Themistoclakis, Chemnitz, September 26-3, 2 4

5 DISCRETE DE LA VALLEE POUSSIN MEANS By applying the previous Gaussian rule to the integrals in () V n,m (w, f, x) = (2) V n,m (w, f, x) = n+m k= H n,m (w, x, y)f(y)w(y)dy µ m n,k c k (w, f)p k (w, x) we obtain the following discrete operator: ( ) Ṽ n,m (w, f, x) = (2 ) Ṽ n,m (w, f, x) = n λ n,j H n,m (w, x, x n,j )f(x n,j ) j= n+m k= µ m n,k c n,k (w, f) p k (w, x) Discrete Fourier coefficients: c n,k (w, f) := n j= λ n,jf(x n,j )p k (w, x n,j ) Woula Themistoclakis, Chemnitz, September 26-3, 2 5

6 COMPARISON WITH LAGRANGE INTERPOLATION L n (w, f, x) = n j= λ n,j K n (w, x, x n,j )f(x n,j ), Ṽ n,m (w, f, x) = n j= λ n,j H n,m (w, x, x n,j )f(x n,j ), m < n Computation: Both computable from the data f(x n,j ), j =,.., n Invariance: { Ln (w, P ) = S n (w, P ) = P, P P n Ṽ n,m (w, P ) = V n,m (w, P ) = P, P P n m Approximation: { L n (w) C u C u C log n Is sup n,m Ṽn,m(w) C u C u for any u < true for suitable u?? Interpolation: { Ln (w, f, x n,j ) = f(x n,j ), j =,..., n Is it true that Ṽn,m(w, f, x n,j ) = f(x n,j )?? Woula Themistoclakis, Chemnitz, September 26-3, 2 6

7 COMPARISON WITH LAGRANGE INTERPOLATION L n (w, f, x) = n j= λ n,j K n (w, x, x n,j )f(x n,j ), Ṽ n,m (w, f, x) = n j= λ n,j H n,m (w, x, x n,j )f(x n,j ), m < n Computation: Both computable from the data f(x n,j ), j =,.., n Invariance: { Ln (w, P ) = S n (w, P ) = P, P P n Ṽ n,m (w, P ) = V n,m (w, P ) = P, P P n m Approximation: { L n (w) C u C u C log n Is sup n,m Ṽn,m(w) C u C u for any u < true for suitable u?? Interpolation: { Ln (w, f, x n,j ) = f(x n,j ), j =,..., n Is it true that Ṽn,m(w, f, x n,j ) = f(x n,j )?? Woula Themistoclakis, Chemnitz, September 26-3, 2 7

8 COMPARISON WITH LAGRANGE INTERPOLATION L n (w, f, x) = n j= λ n,j K n (w, x, x n,j )f(x n,j ), Ṽ n,m (w, f, x) = n j= λ n,j H n,m (w, x, x n,j )f(x n,j ), m < n Computation: Both computable from the data f(x n,j ), j =,.., n Invariance: { Ln (w, P ) = S n (w, P ) = P, P P n Ṽ n,m (w, P ) = V n,m (w, P ) = P, P P n m Approximation: { L n (w) C u C u C log n Is sup n,m Ṽn,m(w) C u C u for any u < true for suitable u?? Interpolation: { Ln (w, f, x n,j ) = f(x n,j ), j =,..., n Is it true that Ṽn,m(w, f, x n,j ) = f(x n,j )?? Woula Themistoclakis, Chemnitz, September 26-3, 2 8

9 COMPARISON WITH LAGRANGE INTERPOLATION L n (w, f, x) = n j= λ n,j K n (w, x, x n,j )f(x n,j ), Ṽ n,m (w, f, x) = n j= λ n,j H n,m (w, x, x n,j )f(x n,j ), m < n Computation: Both computable from the data f(x n,j ), j =,.., n Invariance: { Ln (w, P ) = S n (w, P ) = P, P P n Ṽ n,m (w, P ) = V n,m (w, P ) = P, P P n m Approximation: { Ln (w) C u Cu C log n for any u Is sup n,m Ṽn,m(w) C u Cu < true for suitable u?? Interpolation: { Ln (w, f, x n,j ) = f(x n,j ), j =,..., n Is it true that Ṽn,m(w, f, x n,j ) = f(x n,j )?? Woula Themistoclakis, Chemnitz, September 26-3, 2 9

10 COMPARISON WITH LAGRANGE INTERPOLATION L n (w, f, x) = n j= λ n,j K n (w, x, x n,j )f(x n,j ), Ṽ n,m (w, f, x) = n j= λ n,j H n,m (w, x, x n,j )f(x n,j ), m < n Computation: Both computable from the data f(x n,j ), j =,.., n Invariance: { Ln (w, P ) = S n (w, P ) = P, P P n Ṽ n,m (w, P ) = V n,m (w, P ) = P, P P n m Approximation: { Ln (w) C u Cu C log n for any u Is sup n,m Ṽn,m(w) C u Cu < true for suitable u?? Interpolation: { Ln (w, f, x n,j ) = f(x n,j ), j =,..., n Is it true that Ṽn,m(w, f, x n,j ) = f(x n,j )?? Woula Themistoclakis, Chemnitz, September 26-3, 2

11 Connection with the continuous de la Vallée Poussin operator Case p < : We have Ṽn,m(w, f)u p C V n,m (w) L p w u L p w u where p + p = and λ n(v, x n,k ) := [ n k= ( n k= p 2 k(v, x n,k ) λ n (u p, x n,k ) f(x n,k ) p ) p ] v(x n,k ) x 2 n,k n Case p = : We have Ṽn,m(w, f)u C V n,m (w) C u C u ( ) max f(x n,k) u(x n,k ) k n C > being independent of n, m, f in both the cases. Woula Themistoclakis, Chemnitz, September 26-3, 2

12 Proof for p =. By λ n,k = λ n (w, x n,k ) w(x n,k ) Marcinkiewicz inequality, we get x 2 n,k n, and [ ] n Ṽn,m(w, f)u = max u(x) λ n (w, x n,k )H n,m (w, x, x n,k )f(x n,k ) x k= [ ] n ( w ) C max u(x) λ n x u, x n,k H n,m (w, x, x n,k )(fu)(x n,k ) k= ( ) [ ] n ( w ) C max (fu)(x n,k) u(x) λ n k n u, x n,k H n,m (w, x, x n,k ) C = C ( max k n (fu)(x n,k) ) ( ) max (fu)(x n,k) k n max x max x [ u(x) k= V n,m (w) C u C u. H n,m (w, x, y) w(y) ] u(y) dy Woula Themistoclakis, Chemnitz, September 26-3, 2 2

13 APPROXIMATION IN C u Theorem : Let w = v α,β L [, ] and u = v γ,δ C[, ] satisfy α 2 4 β 2 4 < γ α , < δ β and γ δ α β 2 Then for all integers n and any m = θn with < θ < arbitrarily fixed, the map Ṽn,m(w) : C u C u is uniformly bounded w.r.t. n, m and E n+m (f) u, [f Ṽn,m(w, f)]u CE n m (f) u, holds for every f C u, C > being independent of f, n, m. Woula Themistoclakis, Chemnitz, September 26-3, 2 3

14 Theorem 2: Let w = v α,β (α, β > ) and u = v γ,δ (γ, δ ) satisfy α ν < γ α ν, β ν < δ β ν, for some ν 2 Then for all integers n and m = θn with < θ < arbitrarily fixed, the map Ṽn,m(w) : C u C u is uniformly bounded w.r.t. n, m and E n+m (f) u, [f Ṽn,m(w, f)]u CE n m (f) u, holds for every f C u, C > being independent of f, n, m. Woula Themistoclakis, Chemnitz, September 26-3, 2 4

15 COMPARISON WITH LAGRANGE INTERPOLATION Let w = v α,β L [, ] and u = v γ,δ C[, ] be such that: α β γ α , δ β Then for all sufficiently large pair of integers n and m = θn ( < θ < fixed) and for each f C u, we have Lagrange error: [f L n (w, f)]u C log n E n (f) u, De la V.P. error: [f Ṽ m n (w, f)]u CE n m (f) u, where C > is independent of n, f in both the cases. Woula Themistoclakis, Chemnitz, September 26-3, 2 5

16 APPROXIMATION IN L p u Theorem 3: Let p < and assume that w = v α,β L [, ] and u = v γ,δ L p [, ], with w u Lp [, ], satisfy the bounds α ν < γ + p α ν β ν < δ + p β ν for some ν 2 Then for all n, m N with m = θn ( < θ < fixed) and each f L p u (everywhere defined on ], [), we have Ṽn,m(w, f)u p C ( n k= where we recall that λ n (u p, x n,k ) u p (x n,k ) λ n (u p, x n,k ) f(x n,k ) p ) p x 2 n,k n. Lagrange case L n (w, f): The same estimate holds with p and ν =. Woula Themistoclakis, Chemnitz, September 26-3, 2 6

17 Error estimates in Sobolev type spaces Wr p (u) := { } f L p u : f (r ) AC loc, and f (r) ϕ r L p u f p Wr (u) := fu p + f (r) ϕ r u p, ϕ(x) := x 2 Note: For any f W p r (u), we have E n (f) u,p C n r f (r) ϕ r u p Theorem 4: Under the assumptions of Theorem 3, for all f W p r (u), we have [f Ṽn,m(w, f)]u p C n r f (r) ϕ r u p, f Ṽn,m(w, f) W p s (u) C n r s f W p r (u), < s r where C > is independent of f, n, m and p (setting L u := C u). Lagrange case: L n (w, f) verifies the same estimates, but for p / {, } and ν =. Woula Themistoclakis, Chemnitz, September 26-3, 2 7

18 COMPARISON WITH LAGRANGE INTERPOLATION L n (w, f, x) := Ṽ n,m (w, f, x) := n λ n,k K n (w, x, x n,k )f(x n,k ), k= n λ n,k H n,m (w, x, x n,k )f(x n,k ), k= n > m Invariance: Ṽn,m(w) : f Ṽn,m(w, f) P n+m is a quasi projection Approximation: Ṽn,m(w, f) solves the critical cases p =,. Interpolation: Is it true that Ṽn,m(w, f, x n,k ) = f(x n,k ), k =,.., n?? Woula Themistoclakis, Chemnitz, September 26-3, 2 8

19 Theorem 5 Let w be such that, for all x ], [, we have (3) p n+s (w, x)+p n s (w, x) = p n (w, x)q(x), deg(q) s < n, Then Ṽn,m(w, f, x n,i ) = f(x n,i ), i =,.., n, holds for all n m >. Examples: Bernstein Szego weights defined by w(x) := ( x) α ( + x) β, α = β =, Chebyshev weights 2 w(x) := w(x) :=, w(x) := p(x) x 2 p(x) x2, deg(p) 2 p(x) x + x, deg(p) provide polynomials satisfying (3). Woula Themistoclakis, Chemnitz, September 26-3, 2 9

20 Proof. Note that we can write H n,m (w, x n,i, x n,j ) = 2m where: m r= K n+r (w, x n,i, x n,j ) = K n (w, x n,i, x n,j ) + [ Kn+r (w, x n,i, x n,j ) + K n (r+) (w, x n,i, x n,j ) ] r p n+s (w, x n,i )p n+s (w, x n,j ), + = + + 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 ). Hence by (3) we get p n+s (w, x n,i ) = p n s (w, x n,i ), i =,.., n, and the kernel in n Ṽ 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= s= s= 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 ]. Woula Themistoclakis, Chemnitz, September 26-3, 2 2

21 COMPARISON WITH LAGRANGE INTERPOLATION L n (w, f, x) := Ṽ n,m (w, f, x) := n λ n,k K n (w, x, x n,k )f(x n,k ), k= n λ n,k H n,m (w, x, x n,k )f(x n,k ), k= n > m Invariance: { Ln (w) : f L n (w, f) P n projection Ṽ n,m (w) : f Ṽn,m(w, f) P n+m quasi projection Approximation: { [f Ln (w, f)]u C log n E n (f) u, [f Ṽn,m(w, f)]u CE n m (f) u, Interpolation: { Ln (w, f, x n,k ) = f(x n,k ), for all w = v α,β Ṽ n,m (w, f, x n,k ) = f(x n,k ), α = β = 2, n m Woula Themistoclakis, Chemnitz, September 26-3, 2 2

22 De la Vallée Poussin type polynomial spaces DEF: S n,m (w) := span{ λ n,k H n,m (w, x, x n,k ) : k =,..., 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: Ṽn,m(w, P ) = P, deg(p ) n m P n m S n,m (w) P n+m Theorem 6: In the interpolating case, w = v α,β with α = β = 2, the operator Ṽn,m(w) : f Ṽn,m(w, f) is a projection on S n,m (w), i.e. we have f S n,m (w) f = Ṽn,m(w, f) Woula Themistoclakis, Chemnitz, September 26-3, 2 22

23 INTERPOLATING BASIS OF S n,m (w): { S n,m (w) := span Φ m n,k (w, x) := λ n,k H n,m (w, x, x n,k ), } k =,..., n De la V. P. interpolating polynomial: Ṽ n,m (w, f, x) = n f(x n,k )Φ m n,k(w, x) k= Under the assumptions of Theorem 3, for all a k R, k =,.., n, we have ( ) ( n p n u λ a k Φn,k(w)) m k (u p, x n,k ) a k p if p < k= k= p max a k u(x n,k ) if p = k n i.e. {Φ m n,k (w)} k is a Marcinkiewicz basis in L p u, for all p. Woula Themistoclakis, Chemnitz, September 26-3, 2 23

24 q k (w) := ORTHOGONAL BASIS OF S n,m (w) p k (w) if k n m m + n k 2m p k(w) m n + k 2m p 2n k(w) if n m < k < n 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 =,,..., n } with q h (w, x)q k (w, x)w(x)dx = δ h,k if k n m m 2 +(n k) 2 De la Vallée Poussin interpolating polynomial: [ n n ] Ṽ n,m (w, f, x) = λ n,i p k (w, x n,i )f(x n,i ) q k (w, x) k= i= 2m 2 if n m < k < n Woula Themistoclakis, Chemnitz, September 26-3, 2 24

25 Proof. We are going to state the basis transformation n Φ m n,k(w, x) = λ n,k j= p j (w, x n,k )q j (w, x), k =,..., n. Recall that n+m Φ m n,k(w, s x) := λ n,k H n,m (w, x, x n,k ) = λ n,k µ m n,j p j (w, x n,k )p j (w, x) = λ n,k + n m j= n+m j=n+ n j= n + m j p j (w, x n,k )p j (w, x) + 2m p j(w, x n,k )p j (w, x) j=n m+ n + m j 2m p j(w, x n,k )p j (w, x) i.e., by changing the summation variables, we have Woula Themistoclakis, Chemnitz, September 26-3, 2 25

26 Φ m n,k = λ n,k + m s= n m j= p j (w, x n,k )p j (w) + m s= ] m s 2m p n+s(w, x n,k )p n+s (w) and using p n+s (w, x n,k ) = p n s (w, x n,k ), we get n m Φ m n,k s = λ n,k j= m + λ n,k n = λ n,k j= s= p j (w, x n,k )p j (w) + p n s (w, x n,k ) p j (w, x n,k )q j (w). m + s 2m p n s(w, x n,k )p n s (w) [ m + s 2m p n s(w) m s ] 2m p n+s(w) Woula Themistoclakis, Chemnitz, September 26-3, 2 26

27 COMPARISON WITH LAGRANGE INTERPOLATION Chebyshev case: L n (w, f, x) := Ṽ n,m (w, f, x) := n k= n k= c n,k (w, f) p k (w, x), c n,k (w, f) q k (w, x), n > m Interpolation: { Ln (w, f, x n,k ) = f(x n,k ), k =,..., n Ṽ n,m (w, f, x n,k ) = f(x n,k ), k =,..., n Invariance: { Ln (w) : f L n (w, f) P n projection Ṽ n,m (w) : f Ṽn,m(w, f) S n,m (w) projection Approximation: { [f Ln (w, f)]u C log n E n (f) u, [f Ṽn,m(w, f)]u CE n m (f) u, Woula Themistoclakis, Chemnitz, September 26-3, 2 27

28 .5 Lagrange interpolation for n = Woula Themistoclakis, Chemnitz, September 26-3, 2 28

29 .5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 29

30 .5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 3

31 .5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 3

32 .5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 32

33 .5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 33

34 .5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 34

35 .5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 35

36 .5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 36

37 .5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 37

38 .5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 38

39 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 39

40 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 4

41 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 4

42 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 42

43 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 43

44 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 44

45 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 45

46 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 46

47 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 47

48 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 48

49 .2 Lagrange interpolation for n = Woula Themistoclakis, Chemnitz, September 26-3, 2 49

50 .2 De la V.P. interpolation for n = 2 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 5

51 .2 De la V.P. interpolation for n = 2 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 5

52 .2 De la V.P. interpolation for n = 2 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 52

53 .2 De la V.P. interpolation for n = 2 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 53

54 .2 De la V.P. interpolation for n = 2 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 54

55 .2 De la V.P. interpolation for n = 2 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 55

56 .2 De la V.P. interpolation for n = 2 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 56

57 .2 De la V.P. interpolation for n = 2 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 57

58 .2 De la V.P. interpolation for n = 2 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 58

59 .2 De la V.P. interpolation for n = 2 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 59

60 .4 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 6

61 .4 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 6

62 .4 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 62

63 .4 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 63

64 .4 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 64

65 .4 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 65

66 .4 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 66

67 .4 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 67

68 .4 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 68

69 .4 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 69

70 .2 Fundamental Lagrange polynomial for n = Woula Themistoclakis, Chemnitz, September 26-3, 2 7

71 .2 De la VP kernel polynomials for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 7

72 .2 De la VP kernel polynomials for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 72

73 .2 De la VP kernel polynomials for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 73

74 .2 De la VP kernel polynomials for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 74

75 .2 De la VP kernel polynomials for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 75

76 .2 De la VP kernel polynomials for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 76

77 .2 De la VP kernel polynomials for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 77

78 .2 De la VP kernel polynomials for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 78

79 .2 De la VP kernel polynomials for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 79

80 .2 Fejer kernel polynomials for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 8

81 2.5 Lagrange interpolation for n = Woula Themistoclakis, Chemnitz, September 26-3, 2 8

82 2.5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 82

83 2.5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 83

84 2.5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 84

85 2.5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 85

86 2.5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 86

87 2.5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 87

88 2.5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 88

89 2.5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 89

90 2.5 De la V.P. interpolation for n = 5 and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 9

91 .5 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 9

92 .5 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 92

93 .5 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 93

94 .5 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 94

95 .5 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 95

96 .5 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 96

97 .5 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 97

98 .5 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 98

99 .5 Error curves by Lagrange (in red) and de la V.P. interpolation for θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 99

100 3.5 Lagrange interpolation for n = Woula Themistoclakis, Chemnitz, September 26-3, 2

101 3.5 De la V.P. interpolation for n = and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2

102 3.5 De la V.P. interpolation for n = and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 2

103 3.5 De la V.P. interpolation for n = and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 3

104 3.5 De la V.P. interpolation for n = and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 4

105 3.5 De la V.P. interpolation for n = and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 5

106 3.5 De la V.P. interpolation for n = and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 6

107 3.5 De la V.P. interpolation for n = and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 7

108 3.5 De la V.P. interpolation for n = and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 8

109 3.5 De la V.P. interpolation for n = and m = θ n, with θ = Woula Themistoclakis, Chemnitz, September 26-3, 2 9

110 SOME REFERENCES: M.R.Capobianco, G.Criscuolo, G.Mastroianni, Special Lagrange and Hermite interpolation processes in: Approximation Theory and Applications, T.M.Rassias Editor, Hadronic Press, Palm Harbor, USA, ISBN , 998, pp M.R.Capobianco, W.Themistoclakis, On the boundedness of some de la Vallée Poussin operators, East J. Approx., 7, n. 4 (2), Corrigendum in East J. Approx., 3, n. 2 (27), G.Criscuolo, G.Mastroianni, Fourier and Lagrange operators in some Sobolev type spaces, Acta Sci. Math. (Szeged), 6 (995), G.Mastroianni, G.Milovanovic, Interpolation processes. Basic theory and applications. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 28. G. Mastroianni, M.G.Russo, Lagrange interpolation in weighted Besov spaces, Constr. Approx., 5 (999), G. Mastroianni, W. Themistoclakis, De la Vallée Poussin means and Jackson theorem, Acta Sci. Math. (Szeged), 74 (28), W.Themistoclakis, Some interpolating operators of de la Vallée Poussin type, Acta Math. Hungar. 84 No.3 (999), N.Trefethen, Approximation Theory and Approximation Practice (In preparation) Woula Themistoclakis, Chemnitz, September 26-3, 2