Polynomial approximation via de la Vall ee Poussin means Lecture - - PowerPoint PPT Presentation

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 Computational Applications “Mauro Picone”, Naples, Italy.

Woula Themistoclakis, Chemnitz, September 26-30, 2011 1

De la Vall´ ee Poussin means: Vn,m(w, f, x) := 1 2m

n+m−1

• k=n−m

Sk(w, f, x) where            Sn(w, f, x) :=

n

• k=0

ck(w, f)pk(w, x) Fourier sum ck(w, f) := 1

−1

pk(w, y)f(y)w(y)dy Fourier coefficients w(x) := vα,β(x) = (1−x)α(1+x)β, α, β > −1, being a Jacobi weight and {pj(w, x)}j the corresponding system of orthonormal Jacobi polynomials ◮ Quasi-projection: Vn,m(w, P) = P, ∀P ∈ Pn−m ◮ Near best polynomial: (f − Vn,m(w, f))up ≤ CEn−m(f)u,p holds for any 1 ≤ p ≤ ∞, m = θn (θ ∈]0, 1[ fixed) and suitable u, w.

Woula Themistoclakis, Chemnitz, September 26-30, 2011 2

Other forms: Vn,m(w, f, x) = 1

−1

Hn,m(w, x, y)f(y)w(y)dy (1) Vn,m(w, f, x) =

n+m−1

• k=0

µm

n,k ck(w, f)pk(w, x)

(2) Hn,m(w, x, y) := 1 2m

n+m−1

• r=n−m

Kr(w, x, y) de la Vall´ ee Poussin kernel Kr(w, x, y) :=

r

• j=0

pj(w, x)pj(w, y) Darboux kernel ck(w, f) := 1

−1

pk(w, y)f(y)w(y)dy Fourier coefficients µm

n,k

:=    1 if 0 ≤ k ≤ n − m, n + m − k 2m if n − m < k < n + m.

Woula Themistoclakis, Chemnitz, September 26-30, 2011 3

Lagrange interpolation at the zeros of orthogonal polynomials ◮Take the Fourier sum: Sn−1(w, f, x) := 1

−1

Kn−1(w, x, y)f(y)w(y)dy ◮ Apply the Gaussian rule: 1

−1

g(x)w(x)dx =

n

• j=1

λn,j g(xn,j), g ∈ P2n−1 In this way we obtain the Lagrange polynomial of degree n−1 interpolating f at the n zeros {xn,j}j of pn(w), i.e. Ln(w, f, x) :=

n

• j=1

λn,j Kn−1(w, x, xn,j)f(xn,j) with λn,jKn−1(w, xn,i, xn,j) = δi,j, j = 1, . . . , n.

Woula Themistoclakis, Chemnitz, September 26-30, 2011 4

DISCRETE DE LA VALLEE POUSSIN MEANS By applying the previous Gaussian rule to the integrals in (1) Vn,m(w, f, x) = 1

−1

Hn,m(w, x, y)f(y)w(y)dy (2) Vn,m(w, f, x) =

n+m−1

• k=0

µm

n,k ck(w, f)pk(w, x)

we obtain the following discrete operator: (1′) ˜ Vn,m(w, f, x) =

n

• j=1

λn,j Hn,m(w, x, xn,j)f(xn,j) (2′) ˜ Vn,m(w, f, x) =

n+m−1

• k=0

µm

n,k ˜

cn,k(w, f) pk(w, x) Discrete Fourier coefficients: ˜ cn,k(w, f) := n

j=1 λn,jf(xn,j)pk(w, xn,j)

Woula Themistoclakis, Chemnitz, September 26-30, 2011 5

COMPARISON WITH LAGRANGE INTERPOLATION Ln(w, f, x) = n

j=1 λn,j Kn−1(w, x, xn,j)f(xn,j),

˜ Vn,m(w, f, x) = n

j=1 λn,j Hn,m(w, x, xn,j)f(xn,j),

m < n ◮ Computation: Both computable from the data f(xn,j), j = 1, .., n ◮ Invariance:

• Ln(w, P)

= Sn−1(w, P) = P, ∀P ∈ Pn−1 ˜ Vn,m(w, P) = Vn,m(w, P) = P, ∀P ∈ Pn−m ◮ Approximation:

• Ln(w)C0

u→C0 u ≥ C log n

for any u Is supn,m ˜ Vn,m(w)C0

u→C0 u < ∞

true for suitable u ?? ◮ Interpolation: Ln(w, f, xn,j) = f(xn,j), j = 1, . . . , n Is it true that ˜ Vn,m(w, f, xn,j) = f(xn,j) ??

Woula Themistoclakis, Chemnitz, September 26-30, 2011 6

COMPARISON WITH LAGRANGE INTERPOLATION Ln(w, f, x) = n

j=1 λn,j Kn−1(w, x, xn,j)f(xn,j),

˜ Vn,m(w, f, x) = n

j=1 λn,j Hn,m(w, x, xn,j)f(xn,j),

m < n ◮ Computation: Both computable from the data f(xn,j), j = 1, .., n ◮ Invariance:

• Ln(w, P)

= Sn−1(w, P) = P, ∀P ∈ Pn−1 ˜ Vn,m(w, P) = Vn,m(w, P) = P, ∀P ∈ Pn−m ◮ Approximation:

• Ln(w)C0

u→C0 u ≥ C log n

for any u Is supn,m ˜ Vn,m(w)C0

u→C0 u < ∞

true for suitable u ?? ◮ Interpolation: Ln(w, f, xn,j) = f(xn,j), j = 1, . . . , n Is it true that ˜ Vn,m(w, f, xn,j) = f(xn,j) ??

Woula Themistoclakis, Chemnitz, September 26-30, 2011 7

COMPARISON WITH LAGRANGE INTERPOLATION Ln(w, f, x) = n

j=1 λn,j Kn−1(w, x, xn,j)f(xn,j),

˜ Vn,m(w, f, x) = n

j=1 λn,j Hn,m(w, x, xn,j)f(xn,j),

m < n ◮ Computation: Both computable from the data f(xn,j), j = 1, .., n ◮ Invariance:

• Ln(w, P)

= Sn−1(w, P) = P, ∀P ∈ Pn−1 ˜ Vn,m(w, P) = Vn,m(w, P) = P, ∀P ∈ Pn−m ◮ Approximation:

• Ln(w)C0

u→C0 u ≥ C log n

for any u Is supn,m ˜ Vn,m(w)C0

u→C0 u < ∞

true for suitable u ?? ◮ Interpolation: Ln(w, f, xn,j) = f(xn,j), j = 1, . . . , n Is it true that ˜ Vn,m(w, f, xn,j) = f(xn,j) ??

Woula Themistoclakis, Chemnitz, September 26-30, 2011 8

COMPARISON WITH LAGRANGE INTERPOLATION Ln(w, f, x) = n

j=1 λn,j Kn−1(w, x, xn,j)f(xn,j),

˜ Vn,m(w, f, x) = n

j=1 λn,j Hn,m(w, x, xn,j)f(xn,j),

m < n ◮ Computation: Both computable from the data f(xn,j), j = 1, .., n ◮ Invariance:

• Ln(w, P)

= Sn−1(w, P) = P, ∀P ∈ Pn−1 ˜ Vn,m(w, P) = Vn,m(w, P) = P, ∀P ∈ Pn−m ◮ Approximation: Ln(w)C0

u→C0 u ≥ C log n for any u

Is supn,m ˜ Vn,m(w)C0

u→C0 u < ∞ true for suitable u ??

◮ Interpolation: Ln(w, f, xn,j) = f(xn,j), j = 1, . . . , n Is it true that ˜ Vn,m(w, f, xn,j) = f(xn,j) ??

Woula Themistoclakis, Chemnitz, September 26-30, 2011 9

COMPARISON WITH LAGRANGE INTERPOLATION Ln(w, f, x) = n

j=1 λn,j Kn−1(w, x, xn,j)f(xn,j),

˜ Vn,m(w, f, x) = n

j=1 λn,j Hn,m(w, x, xn,j)f(xn,j),

m < n ◮ Computation: Both computable from the data f(xn,j), j = 1, .., n ◮ Invariance:

• Ln(w, P)

= Sn−1(w, P) = P, ∀P ∈ Pn−1 ˜ Vn,m(w, P) = Vn,m(w, P) = P, ∀P ∈ Pn−m ◮ Approximation: Ln(w)C0

u→C0 u ≥ C log n for any u

Is supn,m ˜ Vn,m(w)C0

u→C0 u < ∞ true for suitable u ??

◮ Interpolation: Ln(w, f, xn,j) = f(xn,j), j = 1, . . . , n Is it true that ˜ Vn,m(w, f, xn,j) = f(xn,j) ??

Woula Themistoclakis, Chemnitz, September 26-30, 2011 10

Connection with the continuous de la Vall´ ee Poussin operator ◮ Case 1 ≤ p < ∞ : We have ˜ Vn,m(w, f)up ≤ CVn,m(w)Lp′

w u

→Lp′

w u

n

• k=1

λn(up, xn,k)|f(xn,k)|p 1

p

where 1 p+ 1 p′ = 1 and λn(v, xn,k) := n−1

• k=0

p2

k(v, xn,k)

−1 ∼ v(xn,k)

• 1 − x2

n,k

n ◮ Case p = ∞ : We have ˜ Vn,m(w, f)u∞ ≤ CVn,m(w)C0

u→C0 u

• max

1≤k≤n |f(xn,k)|u(xn,k)

• C > 0 being independent of n, m, f in both the cases.

Woula Themistoclakis, Chemnitz, September 26-30, 2011 11

Proof for p = ∞. By λn,k = λn(w, xn,k) ∼ w(xn,k)

• 1−x2

n,k

n

, and Marcinkiewicz inequality, we get ˜ Vn,m(w, f)u∞ = max

|x|≤1

• u(x)
• n
• k=1

λn(w, xn,k)Hn,m(w, x, xn,k)f(xn,k)

• ≤

C max

|x|≤1

• u(x)

n

• k=1

λn w u, xn,k

• |Hn,m(w, x, xn,k)(fu)(xn,k)|
• ≤

C

• max

1≤k≤n |(fu)(xn,k)|

• max

|x|≤1

• u(x)

n

• k=1

λn w u, xn,k

• |Hn,m(w, x, xn,k)|
• ≤

C

• max

1≤k≤n |(fu)(xn,k)|

• max

|x|≤1

• u(x)

1

−1

|Hn,m(w, x, y)| w(y) u(y)dy

• =

C

• max

1≤k≤n |(fu)(xn,k)|

• Vn,m(w)C0

u→C0

• u. ✷

Woula Themistoclakis, Chemnitz, September 26-30, 2011 12

◮ APPROXIMATION IN C0

u

Theorem 1: Let w = vα,β ∈ L1[−1, 1] and u = vγ,δ ∈ C[−1, 1] satisfy          α 2 − 1 4 < γ ≤ α 2 + 5 4, β 2 − 1 4 < δ ≤ β 2 + 5 4 and

• γ − δ − α − β

2

• ≤ 1

Then for all integers n and any m = θn with 0 < θ < 1 arbitrarily fixed, the map ˜ Vn,m(w) : C0

u → C0 u is uniformly bounded w.r.t. n, m and

En+m(f)u,∞ ≤ [f − ˜ Vn,m(w, f)]u∞ ≤ CEn−m(f)u,∞ holds for every f ∈ C0

u, C > 0 being independent of f, n, m.

Woula Themistoclakis, Chemnitz, September 26-30, 2011 13

Theorem 2: Let w = vα,β (α, β > −1) and u = vγ,δ (γ, δ ≥ 0) satisfy        α 2 + 1 4 − ν < γ ≤ α 2 + 5 4 − ν, β 2 + 1 4 − ν < δ ≤ β 2 + 5 4 − ν, for some 0 ≤ ν ≤ 1 2 Then for all integers n and m = θn with 0 < θ < 1 arbitrarily fixed, the map ˜ Vn,m(w) : C0

u → C0 u is uniformly bounded w.r.t. n, m and

En+m(f)u,∞ ≤ [f − ˜ Vn,m(w, f)]u∞ ≤ CEn−m(f)u,∞ holds for every f ∈ C0

u, C > 0 being independent of f, n, m.

Woula Themistoclakis, Chemnitz, September 26-30, 2011 14

◮ COMPARISON WITH LAGRANGE INTERPOLATION Let w = vα,β ∈ L1[−1, 1] and u = vγ,δ ∈ C[−1, 1] be such that: α 2 + 1 4 ≤ γ ≤ α 2 + 5 4, β 2 + 1 4 ≤ δ ≤ β 2 + 5 4. Then for all sufficiently large pair of integers n and m = θn (0 < θ < 1 fixed) and for each f ∈ C0

u, we have

Lagrange error: [f − Ln(w, f)]u∞ ≤ C log n En(f)u,∞ De la V.P. error: [f − ˜ V m

n (w, f)]u∞

≤ CEn−m(f)u,∞ where C > 0 is independent of n, f in both the cases.

Woula Themistoclakis, Chemnitz, September 26-30, 2011 15

◮ APPROXIMATION IN Lp

u

Theorem 3: Let 1 ≤ p < ∞ and assume that w = vα,β ∈ L1[−1, 1] and u = vγ,δ ∈ Lp[−1, 1], with w

u ∈ Lp′[−1, 1], satisfy the bounds

       α 2 + 1 4 − ν < γ + 1 p ≤ α 2 + 5 4 − ν β 2 + 1 4 − ν < δ + 1 p ≤ β 2 + 5 4 − ν for some 0 ≤ ν ≤ 1 2 Then for all n, m ∈ N with m = θn (0 < θ < 1 fixed) and each f ∈ Lp

u

(everywhere defined on ] − 1, 1[), we have ˜ Vn,m(w, f)up ≤ C n

• k=1

λn(up, xn,k)|f(xn,k)|p 1

p

where we recall that λn(up, xn,k) ∼ up(xn,k)

• 1−x2

n,k

n

. Lagrange case Ln(w, f): The same estimate holds with p = 1 and ν = 0.

Woula Themistoclakis, Chemnitz, September 26-30, 2011 16

Error estimates in Sobolev–type spaces    W p

r (u)

:=

• f ∈ Lp

u : f (r−1) ∈ ACloc, and f (r)ϕr ∈ Lp u

• fW p

r (u)

:= fup + f (r)ϕrup, ϕ(x) := √ 1 − x2 Note: For any f ∈ W p

r (u), we have En(f)u,p ≤ C nr f (r)ϕrup

Theorem 4: Under the assumptions of Theorem 3, for all f ∈ W p

r (u), we have

[f − ˜ Vn,m(w, f)]up ≤ C nr f (r)ϕrup, f − ˜ Vn,m(w, f)W p

s (u)

≤ C nr−s fW p

r (u),

0 < s ≤ r where C > 0 is independent of f, n, m and 1 ≤ p ≤ ∞ (setting L∞

u := C0 u).

Lagrange case: Ln(w, f) verifies the same estimates, but for p / ∈ {1, ∞} and ν = 0.

Woula Themistoclakis, Chemnitz, September 26-30, 2011 17

COMPARISON WITH LAGRANGE INTERPOLATION Ln(w, f, x) :=

n

• k=1

λn,k Kn−1(w, x, xn,k)f(xn,k), ˜ Vn,m(w, f, x) :=

n

• k=1

λn,k Hn,m(w, x, xn,k)f(xn,k), n > m ◮ Invariance: ˜ Vn,m(w) : f → ˜ Vn,m(w, f) ∈ Pn+m−1 is a quasi–projection ◮ Approximation: ˜ Vn,m(w, f) solves the “critical” cases p = 1, ∞. ◮ Interpolation: Is it true that ˜ Vn,m(w, f, xn,k) = f(xn,k), k = 1, .., n??

Woula Themistoclakis, Chemnitz, September 26-30, 2011 18

Theorem 5 Let w be such that, for all x ∈] − 1, 1[, we have (3) pn+s(w, x)+pn−s(w, x) = pn(w, x)Q(x), deg(Q) ≤ s < n, Then ˜ Vn,m(w, f, xn,i) = f(xn,i), i = 1, .., n, holds for all n ≥ m > 0. Examples: Bernstein–Szego weights defined by w(x) := (1 − x)α(1 + x)β, |α| = |β| = 1 2, Chebyshev weights w(x) := 1 p(x) 1 √ 1 − x2, w(x) := 1 p(x)

• 1 − x

1 + x, deg(p) ≤ 1 w(x) := 1 p(x)

• 1 − x2,

deg(p) ≤ 2 provide polynomials satisfying (3).

Woula Themistoclakis, Chemnitz, September 26-30, 2011 19

• Proof. Note that we can write

Hn,m(w, xn,i, xn,j) = 1 2m

m−1

• r=0
• Kn+r(w, xn,i, xn,j) + Kn−(r+1)(w, xn,i, xn,j)
• where:

Kn+r(w, xn,i, xn,j) = Kn(w, xn,i, xn,j) +

r

• s=1

pn+s(w, xn,i)pn+s(w, xn,j), + = + + Kn−(r+1)(w, xn,i, xn,j) = Kn(w, xn,i, xn,j) −

r

• s=1

pn−s(w, xn,i)pn−s(w, xn,j). Hence by (3) we get pn+s(w, xn,i) = −pn−s(w, xn,i), i = 1, .., n, and the kernel in ˜ Vn,m(w, f, xn,i) =

n

• j=1

λn,jHn,m(w, xn,i, xn,j)f(xn,j) reduces to Hn,m(w, xn,i, xn,j) = Kn(w, xn,i, xn,j) = δi,j[λn,j]−1. ✷

Woula Themistoclakis, Chemnitz, September 26-30, 2011 20

COMPARISON WITH LAGRANGE INTERPOLATION Ln(w, f, x) :=

n

• k=1

λn,k Kn−1(w, x, xn,k)f(xn,k), ˜ Vn,m(w, f, x) :=

n

• k=1

λn,k Hn,m(w, x, xn,k)f(xn,k), n > m ◮ Invariance:

• Ln(w)

: f → Ln(w, f) ∈ Pn−1 projection ˜ Vn,m(w) : f → ˜ Vn,m(w, f) ∈ Pn+m−1 quasi–projection ◮ Approximation:

• [f − Ln(w, f)]u∞

≤ C log n En(f)u,∞ [f − ˜ Vn,m(w, f)]u∞ ≤ CEn−m(f)u,∞ ◮ Interpolation:

• Ln(w, f, xn,k) = f(xn,k),

for all w = vα,β ˜ Vn,m(w, f, xn,k) = f(xn,k), |α| = |β| = 1

2, n ≥ m

Woula Themistoclakis, Chemnitz, September 26-30, 2011 21

De la Vall´ ee Poussin type polynomial spaces DEF: Sn,m(w) := span{ λn,k Hn,m(w, x, xn,k) : k = 1, . . . , n} ◮ Interpolation property: λn,k Hn,m(w, xn,h, xn,k) = δh,k ⇓ dim Sn,m(w) = n ◮ Invariance property: ˜ Vn,m(w, P) = P, deg(P) ≤ n − m ⇓ Pn−m ⊂ Sn,m(w) ⊂ Pn+m−1 Theorem 6: In the interpolating case, w = vα,β with |α| = |β| = 1

2, the

• perator ˜

Vn,m(w) : f → ˜ Vn,m(w, f) is a projection on Sn,m(w), i.e. we have f ∈ Sn,m(w) ⇔ f = ˜ Vn,m(w, f)

Woula Themistoclakis, Chemnitz, September 26-30, 2011 22

INTERPOLATING BASIS OF Sn,m(w): Sn,m(w) := span

• Φm

n,k(w, x) := λn,k Hn,m(w, x, xn,k),

k = 1, . . . , n

• De la V. P. interpolating polynomial:

˜ Vn,m(w, f, x) =

n

• k=1

f(xn,k)Φm

n,k(w, x)

Under the assumptions of Theorem 3, for all ak ∈ R, k = 1, .., n, we have

• u

n

• k=1

akΦm

n,k(w)

• p

∼          n

• k=1

λk(up, xn,k)|ak|p 1

p

if 1 ≤ p < ∞ max

1≤k≤n |ak|u(xn,k)

if p = ∞ i.e. {Φm

n,k(w)}k is a Marcinkiewicz basis in Lp u, for all 1 ≤ p ≤ ∞.

Woula Themistoclakis, Chemnitz, September 26-30, 2011 23

ORTHOGONAL BASIS OF Sn,m(w) qk(w) :=    pk(w) if 0 ≤ k ≤ n − m m + n − k 2m pk(w) − m − n + k 2m p2n−k(w) if n − m < k < n Theorem 7: The set {qk(w)}k is an orthogonal basis of Sn,m(w), i.e. we have Sn,m(w) := span{qk(w) : k = 0, 1, . . . , n − 1} with 1

−1

qh(w, x)qk(w, x)w(x)dx = δh,k·    1 if 0 ≤ k ≤ n−m m2+(n−k)2 2m2 if n−m < k < n De la Vall´ ee Poussin interpolating polynomial: ˜ Vn,m(w, f, x) =

n−1

• k=0

n

• i=1

λn,ipk(w, xn,i)f(xn,i)

• qk(w, x)

Woula Themistoclakis, Chemnitz, September 26-30, 2011 24

• Proof. We are going to state the basis transformation

Φm

n,k(w, x) = λn,k n−1

• j=0

pj(w, xn,k)qj(w, x), k = 1, . . . , n. Recall that s Φm

n,k(w, x) := λn,k Hn,m(w, x, xn,k) = λn,k n+m−1

• j=0

µm

n,j pj(w, xn,k)pj(w, x)

= λn,k  

n−m

• j=0

pj(w, xn,k)pj(w, x) +

n−1

• j=n−m+1

n + m − j 2m pj(w, xn,k)pj(w, x) +

n+m−1

• j=n+1

n + m − j 2m pj(w, xn,k)pj(w, x)   i.e., by changing the summation variables, we have

Woula Themistoclakis, Chemnitz, September 26-30, 2011 25

Φm

n,k

= λn,k  

n−m

• j=0

pj(w, xn,k)pj(w) +

m−1

• s=1

m + s 2m pn−s(w, xn,k)pn−s(w) +

m−1

• s=1

m − s 2m pn+s(w, xn,k)pn+s(w)

• and using pn+s(w, xn,k) = −pn−s(w, xn,k), we get

s Φm

n,k

= λn,k

n−m

• j=0

pj(w, xn,k)pj(w) + + λn,k

m−1

• s=1

pn−s(w, xn,k) m + s 2m pn−s(w) − m − s 2m pn+s(w)

• =

λn,k

n−1

• j=0

pj(w, xn,k)qj(w). ✷

Woula Themistoclakis, Chemnitz, September 26-30, 2011 26

COMPARISON WITH LAGRANGE INTERPOLATION Chebyshev case: Ln(w, f, x) :=

n−1

• k=0

˜ cn,k(w, f) pk(w, x), ˜ Vn,m(w, f, x) :=

n−1

• k=0

˜ cn,k(w, f) qk(w, x), n > m ◮ Interpolation:

• Ln(w, f, xn,k)

= f(xn,k), k = 1, . . . , n ˜ Vn,m(w, f, xn,k) = f(xn,k), k = 1, . . . , n ◮ Invariance:

• Ln(w)

: f → Ln(w, f) ∈ Pn−1 projection ˜ Vn,m(w) : f → ˜ Vn,m(w, f) ∈ Sn,m(w) projection ◮ Approximation:

• [f − Ln(w, f)]u∞

≤ C log n En(f)u,∞ [f − ˜ Vn,m(w, f)]u∞ ≤ CEn−m(f)u,∞

Woula Themistoclakis, Chemnitz, September 26-30, 2011 27

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

−1 −0.5 0.5 1 −0.5 0.5 1 1.5 2 2.5 3 3.5 De la V.P. interpolation for n = 100 and m = θ n, with θ = 0.1 Woula Themistoclakis, Chemnitz, September 26-30, 2011 101

−1 −0.5 0.5 1 −0.5 0.5 1 1.5 2 2.5 3 3.5 De la V.P. interpolation for n = 100 and m = θ n, with θ = 0.2 Woula Themistoclakis, Chemnitz, September 26-30, 2011 102

−1 −0.5 0.5 1 −0.5 0.5 1 1.5 2 2.5 3 3.5 De la V.P. interpolation for n = 100 and m = θ n, with θ = 0.3 Woula Themistoclakis, Chemnitz, September 26-30, 2011 103

−1 −0.5 0.5 1 −0.5 0.5 1 1.5 2 2.5 3 3.5 De la V.P. interpolation for n = 100 and m = θ n, with θ = 0.4 Woula Themistoclakis, Chemnitz, September 26-30, 2011 104

−1 −0.5 0.5 1 −0.5 0.5 1 1.5 2 2.5 3 3.5 De la V.P. interpolation for n = 100 and m = θ n, with θ = 0.5 Woula Themistoclakis, Chemnitz, September 26-30, 2011 105

−1 −0.5 0.5 1 −0.5 0.5 1 1.5 2 2.5 3 3.5 De la V.P. interpolation for n = 100 and m = θ n, with θ = 0.6 Woula Themistoclakis, Chemnitz, September 26-30, 2011 106

−1 −0.5 0.5 1 −0.5 0.5 1 1.5 2 2.5 3 3.5 De la V.P. interpolation for n = 100 and m = θ n, with θ = 0.7 Woula Themistoclakis, Chemnitz, September 26-30, 2011 107

−1 −0.5 0.5 1 −0.5 0.5 1 1.5 2 2.5 3 3.5 De la V.P. interpolation for n = 100 and m = θ n, with θ = 0.8 Woula Themistoclakis, Chemnitz, September 26-30, 2011 108

−1 −0.5 0.5 1 −0.5 0.5 1 1.5 2 2.5 3 3.5 De la V.P. interpolation for n = 100 and m = θ n, with θ = 0.9 Woula Themistoclakis, Chemnitz, September 26-30, 2011 109

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 1-57485-041-5, 1998, pp.37–62

• M.R.Capobianco, W.Themistoclakis, On the boundedness of some de la Vall´

ee Poussin operators, East J. Approx., 7, n. 4 (2001), 417–444. Corrigendum in East

• J. Approx., 13, n. 2 (2007), 223–226.
• G.Criscuolo, G.Mastroianni, Fourier and Lagrange operators in some Sobolev–type

spaces, Acta Sci. Math. (Szeged), 60 (1995), 131–148.

• G.Mastroianni,

G.Milovanovic, Interpolation processes. Basic theory and

• applications. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2008.
• G. Mastroianni, M.G.Russo, Lagrange interpolation in weighted Besov spaces,
• Constr. Approx., 15 (1999), 257–289.
• G. Mastroianni, W. Themistoclakis, De la Vall´

ee Poussin means and Jackson theorem, Acta Sci. Math. (Szeged), 74 (2008), 147–170.

• W.Themistoclakis, Some interpolating operators of de la Vall´

ee Poussin type, Acta Math. Hungar. 84 No.3 (1999), 221-235

• N.Trefethen, Approximation Theory and Approximation Practice (In preparation)

http://www2.maths.ox.ac.uk/chebfun/ATAP/

Woula Themistoclakis, Chemnitz, September 26-30, 2011 110