12. For € [1,1], define the polynomials, Po(x)=1, P₁(x) = x, P₂(x) = 3x²1, P3(x) = 5x³ - 3x. (a) Show that these four functions are orthogonal with respect to the inner product (f,g) = [ f(x)g(x) dx. (b) If f(x) = ao+α₁x + a₂x² + α³x³ +...+ a² is a nontrivial polynomial of degree and f is orthogonal to each of Po, P₁, P2, P3, show that the degree n ≥ 4. [Hints: Assume for a contradiction that n ≤ 3, so f(x) = ao + a₁x + a₂x² + a3x³, and Ow ao = 0= a1 = 0₂ az (and so f(x) is the zero polynomial).] (c) Find a polynomial P4(x) which is of order 4 and which is orthogonal to each of P₁, P2, P3.
12. For € [1,1], define the polynomials, Po(x)=1, P₁(x) = x, P₂(x) = 3x²1, P3(x) = 5x³ - 3x. (a) Show that these four functions are orthogonal with respect to the inner product (f,g) = [ f(x)g(x) dx. (b) If f(x) = ao+α₁x + a₂x² + α³x³ +...+ a² is a nontrivial polynomial of degree and f is orthogonal to each of Po, P₁, P2, P3, show that the degree n ≥ 4. [Hints: Assume for a contradiction that n ≤ 3, so f(x) = ao + a₁x + a₂x² + a3x³, and Ow ao = 0= a1 = 0₂ az (and so f(x) is the zero polynomial).] (c) Find a polynomial P4(x) which is of order 4 and which is orthogonal to each of P₁, P2, P3.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please show a step-by-step solution. Do not skip steps, and explain your steps. Write it on paper, preferably. Make sure the work is clear.
![12. For x € [-1, 1], define the polynomials,
Po(x)=1, P₁(x) = x,
P₂(x) = 3x²1, P3(x) = 5x³ - 3x.
(a) Show that these four functions are orthogonal with respect to the inner product
(f,g) = [ f(x)g(x) dr.
(b) If f(x) = a + α₁x + a₂x² + a²x³ + ... + ax is a nontrivial polynomial of degree
n, and f is orthogonal to each of Po, P₁, P2, P3, show that the degree n ≥ 4.
[Hints: Assume for a contradiction that n ≤ 3, so f(x) = ao + a₁x + a₂x² + a3x³, and
show ao = 0 = α₁ = a2 = a3 (and so f(x) is the zero polynomial).]
(c) Find a polynomial P(x) which is of order 4 and which is orthogonal to each of
Po, P1, P2, P3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cb378ba-be87-4838-bd31-84fa6b2aaf1f%2F8d0f8fab-017f-42c9-9486-ac6e13f31f54%2Fhkulv4s_processed.png&w=3840&q=75)
Transcribed Image Text:12. For x € [-1, 1], define the polynomials,
Po(x)=1, P₁(x) = x,
P₂(x) = 3x²1, P3(x) = 5x³ - 3x.
(a) Show that these four functions are orthogonal with respect to the inner product
(f,g) = [ f(x)g(x) dr.
(b) If f(x) = a + α₁x + a₂x² + a²x³ + ... + ax is a nontrivial polynomial of degree
n, and f is orthogonal to each of Po, P₁, P2, P3, show that the degree n ≥ 4.
[Hints: Assume for a contradiction that n ≤ 3, so f(x) = ao + a₁x + a₂x² + a3x³, and
show ao = 0 = α₁ = a2 = a3 (and so f(x) is the zero polynomial).]
(c) Find a polynomial P(x) which is of order 4 and which is orthogonal to each of
Po, P1, P2, P3.
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