(b) Explain that function f(x) = 1 is not perpendicular to sin(x) or to sin(3x), but is perpendicular to sin(2x) and sin(4x).
(b) Explain that function f(x) = 1 is not perpendicular to sin(x) or to sin(3x), but is perpendicular to sin(2x) and sin(4x).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
Solve b part only in 30 min and take a thumb up
![Consider the space of differentiable functions f: [0, 1] - R with the L 2 -inner product
(f. g) = 5n 0 f(x)g(x) dx.
(a)
Show that the collection consisting of the features
sin(x),sin(2x), ...,sin( (k - 1)x), (8.1)
defined for x € (0, n), are orthogonal for this inner product.
(b)
Explain that function f(x) = 1 is not perpendicular to sin(x) or to sin(3x), but is perpendicular
to sin(2x) and sin(4x).
(c)
Determine the projection f(x) = 1 along sin(x), sin(3x)and find from there a function of the
form 1 + c1 sin(x) + c3 sin(3x), which is perpendicular to both sin(x) and sin(3x). Explain why
this combination is perpendicular to all functions in (8.1) for k = 5.
(d)
For k = 5, determine in python the linear combination of functions in (8.1), which is closest to
the function
f(x) = 1 - e -x for 0< x <n.
Plot the function f(x) and its approximation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a6958d8-9c7a-418f-aba3-699187d345d6%2F79bbfe7b-54fd-42ac-bfd1-49b1e961ea19%2Ffg19sv7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the space of differentiable functions f: [0, 1] - R with the L 2 -inner product
(f. g) = 5n 0 f(x)g(x) dx.
(a)
Show that the collection consisting of the features
sin(x),sin(2x), ...,sin( (k - 1)x), (8.1)
defined for x € (0, n), are orthogonal for this inner product.
(b)
Explain that function f(x) = 1 is not perpendicular to sin(x) or to sin(3x), but is perpendicular
to sin(2x) and sin(4x).
(c)
Determine the projection f(x) = 1 along sin(x), sin(3x)and find from there a function of the
form 1 + c1 sin(x) + c3 sin(3x), which is perpendicular to both sin(x) and sin(3x). Explain why
this combination is perpendicular to all functions in (8.1) for k = 5.
(d)
For k = 5, determine in python the linear combination of functions in (8.1), which is closest to
the function
f(x) = 1 - e -x for 0< x <n.
Plot the function f(x) and its approximation.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
