4. (a) Let the inner product of piecewise continuous functions on the interval 0 ≤ x ≤ 3 be given by 3 (f; g) = f³ f(x)g(x) dr. L Consider three piecewise constant functions 01(x), 02(x) and 03(x): (2) Ok(x) = { 1, - if k − 1 < x < k : k = 1, 2, 3. 0, " at all other points, i. Show that 1(x), 02(x), 03(x) form an orthonormal system of functions. ii. Find the coefficients a1, a2, a3 for the best in the mean approximation of x² on the interval [0,3]. 3 f(x) = Σ akok (x) k=1 iii. On the same plot sketch graphs of the functions f(x) and x2 for 0 ≤ x ≤ 3.
4. (a) Let the inner product of piecewise continuous functions on the interval 0 ≤ x ≤ 3 be given by 3 (f; g) = f³ f(x)g(x) dr. L Consider three piecewise constant functions 01(x), 02(x) and 03(x): (2) Ok(x) = { 1, - if k − 1 < x < k : k = 1, 2, 3. 0, " at all other points, i. Show that 1(x), 02(x), 03(x) form an orthonormal system of functions. ii. Find the coefficients a1, a2, a3 for the best in the mean approximation of x² on the interval [0,3]. 3 f(x) = Σ akok (x) k=1 iii. On the same plot sketch graphs of the functions f(x) and x2 for 0 ≤ x ≤ 3.
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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![4. (a) Let the inner product of piecewise continuous functions on the interval 0 ≤ x ≤ 3
be given by
3
(f; g) = f³ f(x)g(x) dr.
L
Consider three piecewise constant functions 01(x), 02(x) and 03(x):
(2)
Ok(x) = {
1,
-
if k − 1 < x < k
:
k = 1, 2, 3.
0,
"
at all other points,
i. Show that 1(x), 02(x), 03(x) form an orthonormal system of functions.
ii. Find the coefficients a1, a2, a3 for the best in the mean approximation
of x² on the interval [0,3].
3
f(x) = Σ akok (x)
k=1
iii. On the same plot sketch graphs of the functions f(x) and x2 for 0 ≤ x ≤ 3.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69cce4ac-4bf6-4e6b-8636-bf160e045b58%2F49a463f1-4c71-4ff1-b560-66d0c7dcdb29%2Fz8ya6pq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. (a) Let the inner product of piecewise continuous functions on the interval 0 ≤ x ≤ 3
be given by
3
(f; g) = f³ f(x)g(x) dr.
L
Consider three piecewise constant functions 01(x), 02(x) and 03(x):
(2)
Ok(x) = {
1,
-
if k − 1 < x < k
:
k = 1, 2, 3.
0,
"
at all other points,
i. Show that 1(x), 02(x), 03(x) form an orthonormal system of functions.
ii. Find the coefficients a1, a2, a3 for the best in the mean approximation
of x² on the interval [0,3].
3
f(x) = Σ akok (x)
k=1
iii. On the same plot sketch graphs of the functions f(x) and x2 for 0 ≤ x ≤ 3.
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