Description

• Write in a neat and legible handwriting or use LATEX
• Clearly explain the reasoning process
• Write in a complete style (subject, verb, and object)
• Be critical on your results
Questions preceded by a * are optional. Although they can be skipped without any deduction, it is important to know and understand the results they contain.
Ex. 1 — Lebesgue constant for Chebyshev nodes
In the lectures (slide 3.23) we showed that the choice of the nodes is of a major importance when performing interpolation. This in fact confirmed the observations from lab 1, where for instance Runge’s phenomenon appears as the number of equidistance nodes increases. In this exercise we show that interpolation on Chebyshev nodes is a good choice as in this case the Lebesgue constant grows slowly.
We recall that the Lebesgue number is defined by Λnand the Chebyshev polynomials by Tn(x) = cos(n arccosx). The roots xi, 0 , of Chebyshev polynomials are given by
xi = cos θi, with θi = 2(2ni+1+1)π.
1. Let ℓi be the Lagrange polynomials associated to the node xi.
a) Prove that
Tn+1(x) ℓi(x) = .
(x − xi)Tn′+1(xi)
n + 1 n + 1
b) Show that Tn′+1(x) = √1 − cos2 θ sin(n + 1)θ, and Tn′+1(xk) = (−1)k sinθk .
c) Conclude that
2. We want to show that
* a) Prove that
b) Show that
n
1 θ
.
=0 i=0
cott dt. (1.1)
n
i=0 2
cott dt .
n n
cott dt. 2(n
k=0 k=0 2
c) Prove that equation (1.1) is true.
* 3. Using question 2, conclude that .
Ex. 2 — Interpolation
In this exercise we construct an interpolation method for an abitrary continuous function over [a,b] ⊂ R.
Let C[a,b], a,b ∈ R, be the set of the continuous functions over [a,b], endowed with the usual norm for uniform convergence, ∥u∥∞ = maxx∈[a,b] |u(x)|. For some n ∈ N we define the collection of points (xk,yk), k ∈ {0,··· ,n}, such that a ≤ x0 < y0 < x1 < y1 < ··· < xn < yn ≤ b and consider the following application
Φ : C[a,b] −→ Rn+1 f 7−→ (m0(f ),m1(f ),··· ,mn(f )),
with for all k ∈ {0,··· ,n}, mk . In other words the application Φ maps a function defined on [a,b] onto an element of Rn+1.
1. Let f ∈ C[a,b] such that Φ(f ) = 0. Show that, for any k there exists ξk ∈ [xk,yk] such that
f (ξk) = 0.
2. Prove that if Φ is restricted to Rn[x], then Φ is injective. Conclude on the existence of a unique polynomial Pn ∈ Rn[x] such that Φ(Pn) = Φ(f ).
3. Assuming f ∈ Cn+1[a,b], prove that Pn is the interpolation polynomial of f and conclude that
( )n+1 (n+1)
∥f|f (x)|.
Ex. 3 — Trigonometric polynomials
Let x ∈ [0,1] and θ ∈ [−π,pi]. For n ∈ N, we denote by Tn = {Qn,Qn , the set of the trigonometric polynomials of degree less than n.
* 1. Prove that for any 0 ≤ k ≤ n, (cosθ)k is in Tn. Conclude that Φ, which maps Pn(x) into Qn(θ) = Pn(cosθ) is a linear bijection from Rn[x] into Tn.
2. For f ∈ Cn+1[−1,1], we define F(θ) = f (cosθ). Show the existence of Qn ∈ Tn, such that
Qn(θi) = F(θi), where .
3. Prove that finding Qn ∈ Tn in the previous question is equivalent to solving the linear system La = b, with a = (a0,··· ,an) such that Qn(θ) = √0 + ∑kn=1 ak coskθ. a
T
2
4. Show that for any θ ∈ (−π,π), there exists ξ ∈ (−1,1), such that
F f (n+1)(ξ).