In the inner product space C[-1, 1], consider the orthogonal set of polynomials {u₁, U₂, U3}, where x2² - 1/1/1 3 u₁ = 1, Let v = f(x) be the function defined by + U₂ = x, f(x) = - 0 3x x + and U3 = (a) Compute the following inner products. (u₁, ₁) = (U₂, U₂) = (U3, U3) = (u₁, v) = (u₂, v) = (U3, v) = (b) Find the second degree generalized Fourier approximation f₂(x) of f(x). f₂(x) = if x < 0, if x ≥ 0. (x² - 1).
In the inner product space C[-1, 1], consider the orthogonal set of polynomials {u₁, U₂, U3}, where x2² - 1/1/1 3 u₁ = 1, Let v = f(x) be the function defined by + U₂ = x, f(x) = - 0 3x x + and U3 = (a) Compute the following inner products. (u₁, ₁) = (U₂, U₂) = (U3, U3) = (u₁, v) = (u₂, v) = (U3, v) = (b) Find the second degree generalized Fourier approximation f₂(x) of f(x). f₂(x) = if x < 0, if x ≥ 0. (x² - 1).
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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Question
![In the inner product space C[-1, 1], consider the orthogonal set of polynomials {u₁, U₂, U3 }, where
Let v = f(x) be the function defined by
(a) Compute the following inner products.
(u₁, u₁ ) :
=
(U₂, U₂)
(U3, U3) =
(u₁, v) =
(u₂, v) =
=
U₁ =
=
+
1,
U₂ = x,
f(x) = {
0
3x
x +
and
(U3, v)
(b) Find the second degree generalized Fourier approximation f₂(x) of f(x).
f₂(x) =
2
U3 = x²
if x < 0,
if x ≥ 0.
(x²
-im
3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9ff5cc5-ffa0-4d62-80cc-4b5d73842eb0%2Ffb9044d6-24b5-4e38-bad2-ea81470b5b6f%2Fmoo3dih_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In the inner product space C[-1, 1], consider the orthogonal set of polynomials {u₁, U₂, U3 }, where
Let v = f(x) be the function defined by
(a) Compute the following inner products.
(u₁, u₁ ) :
=
(U₂, U₂)
(U3, U3) =
(u₁, v) =
(u₂, v) =
=
U₁ =
=
+
1,
U₂ = x,
f(x) = {
0
3x
x +
and
(U3, v)
(b) Find the second degree generalized Fourier approximation f₂(x) of f(x).
f₂(x) =
2
U3 = x²
if x < 0,
if x ≥ 0.
(x²
-im
3
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