Problem 5. Let n ≥ 0 be an integer, let [a, b] = [−1,1], and let x; be the roots of Tn+1. Recall that deg Tn+1 = n + 1, so Tn+1 has n + 1 zeros. Consider the function: n πn(x) = (x − x₁). i=0 (3) An important fact is that n(x) = Tn+1(x)/2". 1. Define the minimax polynomial interpolation problem. 2. Explain why this important fact means that Chebyshev polynomials are near-optimal for minimax polynomial interpolation. 3. For a function g, let ||9||[a,b] = maxa
Problem 5. Let n ≥ 0 be an integer, let [a, b] = [−1,1], and let x; be the roots of Tn+1. Recall that deg Tn+1 = n + 1, so Tn+1 has n + 1 zeros. Consider the function: n πn(x) = (x − x₁). i=0 (3) An important fact is that n(x) = Tn+1(x)/2". 1. Define the minimax polynomial interpolation problem. 2. Explain why this important fact means that Chebyshev polynomials are near-optimal for minimax polynomial interpolation. 3. For a function g, let ||9||[a,b] = maxa
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![Problem 5. Let n ≥ 0 be an integer, let [a, b] [−1,1], and let x; be the roots of Tn+1. Recall that
deg Tn+1 = n + 1, so Tn+1 has n + 1 zeros. Consider the function:
=
πn(x) =
n
=
= II(x − x₂).
i=0
=
An important fact is that n(x) = Tn+1(x)/2".
1. Define the minimax polynomial interpolation problem.
2. Explain why this important fact means that Chebyshev polynomials are near-optimal for minimax
polynomial interpolation.
3. For a function g, let ||9||[a,b]
maxa≤x≤b|g(x)|. Let ƒ = C(n+¹)([a, b]). Prove that if pn is the degree
n Lagrange polynomial interpolating på with interpolation nodes given by the zeros of Tn+1, then:
(3)
||ƒ − Pn||[a,b] ≤
1
2n(n + 1)! || f(n+¹) ||[a,b].
(4)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F725ee8ea-573b-4075-8ffe-8d3f36956d72%2F2c4d71c8-2669-4b15-9f84-589f06e8f338%2Fthind6f_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 5. Let n ≥ 0 be an integer, let [a, b] [−1,1], and let x; be the roots of Tn+1. Recall that
deg Tn+1 = n + 1, so Tn+1 has n + 1 zeros. Consider the function:
=
πn(x) =
n
=
= II(x − x₂).
i=0
=
An important fact is that n(x) = Tn+1(x)/2".
1. Define the minimax polynomial interpolation problem.
2. Explain why this important fact means that Chebyshev polynomials are near-optimal for minimax
polynomial interpolation.
3. For a function g, let ||9||[a,b]
maxa≤x≤b|g(x)|. Let ƒ = C(n+¹)([a, b]). Prove that if pn is the degree
n Lagrange polynomial interpolating på with interpolation nodes given by the zeros of Tn+1, then:
(3)
||ƒ − Pn||[a,b] ≤
1
2n(n + 1)! || f(n+¹) ||[a,b].
(4)
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