Let f (x) be the 4-periodic function R → R determined for -2 < x< 2 by = 1 for – 1 < x < 1, f(x) : f(x) = 0 for 1 < |x| < 2. a) Calculate by integration the coefficients an and b, in the function f belonging to the real Fourier series, f (x) = ao + E (an Cos(n-r) + bn sin(n n=1 Write up the Fourier series. b) Determine the convergence conditions for the Fourier series, including whether it is uniformly convergent on the interval [-2,2]. For which values of x is the Fourier series convergent, and what does it converge to for such x? Specify specially the Fourier series sum S for x = -1. c) Determine if the function g(x) = cos (2tx) is orthogonal on f (x) in the space L² ([-2,2]), that is equipped with the usual inner product (u | v) = / u(x)v(æ) dæ. -2 Let Vy be the subspace of L? ([-2,2]) given by VN = span{1, cos(,*), sin(,x),..., cos(N,x), sin(N,x)}. ,cos(Nx), sin(N,x)}. Investigate whether g(x) belongs to the orthogonal complement V for N = 5. Justify the answer.
Let f (x) be the 4-periodic function R → R determined for -2 < x< 2 by = 1 for – 1 < x < 1, f(x) : f(x) = 0 for 1 < |x| < 2. a) Calculate by integration the coefficients an and b, in the function f belonging to the real Fourier series, f (x) = ao + E (an Cos(n-r) + bn sin(n n=1 Write up the Fourier series. b) Determine the convergence conditions for the Fourier series, including whether it is uniformly convergent on the interval [-2,2]. For which values of x is the Fourier series convergent, and what does it converge to for such x? Specify specially the Fourier series sum S for x = -1. c) Determine if the function g(x) = cos (2tx) is orthogonal on f (x) in the space L² ([-2,2]), that is equipped with the usual inner product (u | v) = / u(x)v(æ) dæ. -2 Let Vy be the subspace of L? ([-2,2]) given by VN = span{1, cos(,*), sin(,x),..., cos(N,x), sin(N,x)}. ,cos(Nx), sin(N,x)}. Investigate whether g(x) belongs to the orthogonal complement V for N = 5. Justify the answer.
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
Concept explainers
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
Topic Video
Question
I need help with question c. I have attached a picture of the whole assignment.
![Let f (x) be the 4-periodic function R → R determined for -2 <x<2 by
f (x) = 1 for –1< x < 1,
f (x) = 0 for 1 < |x| < 2.
a) Calculate by integration the coefficients an and b, in the function f belonging to the real
Fourier series,
f(x) = ao + (an cos(nx) + b, sin(n-x)).
n=1
Write up the Fourier series.
b) Determine the convergence conditions for the Fourier series, including whether it is
uniformly convergent on the interval [-2,2].
For which values of x is the Fourier series convergent, and what does it converge to for
such x?
Specify specially the Fourier series sum S for x =-1.
c) Determine if the function g(x) = cos (2nx) is orthogonal on f(x) in the space
L² ([-2,2]), that is equipped with the usual inner product
c2
(u | v) = / u(x)v(x) dx.
-2
Let Vy be the subspace of L2 ([-2,2]) given by
VN = span{1, cos(x), sin(x),
..., cos(Ne), sin(N,a)}.
Investigate whether g(x) belongs to the orthogonal complement V for N = 5. Justify the
answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faa3bc07d-af1c-413c-8f75-3b997528cb20%2F5bc3d404-5a48-403d-bd02-dc930155fba5%2F9ve5sna_processed.png&w=3840&q=75)
Transcribed Image Text:Let f (x) be the 4-periodic function R → R determined for -2 <x<2 by
f (x) = 1 for –1< x < 1,
f (x) = 0 for 1 < |x| < 2.
a) Calculate by integration the coefficients an and b, in the function f belonging to the real
Fourier series,
f(x) = ao + (an cos(nx) + b, sin(n-x)).
n=1
Write up the Fourier series.
b) Determine the convergence conditions for the Fourier series, including whether it is
uniformly convergent on the interval [-2,2].
For which values of x is the Fourier series convergent, and what does it converge to for
such x?
Specify specially the Fourier series sum S for x =-1.
c) Determine if the function g(x) = cos (2nx) is orthogonal on f(x) in the space
L² ([-2,2]), that is equipped with the usual inner product
c2
(u | v) = / u(x)v(x) dx.
-2
Let Vy be the subspace of L2 ([-2,2]) given by
VN = span{1, cos(x), sin(x),
..., cos(Ne), sin(N,a)}.
Investigate whether g(x) belongs to the orthogonal complement V for N = 5. Justify the
answer.
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
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.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,
