A periodic function is defined by f(x + p) = f(x), where p is the period of f(x). Assume a periodic function is with period 27 and the inner product of 2 periodic functions is defined as 1 2π (1₁8) = = = = √ ₁²" 1 f.gdt. 2π Jo (a) Please verify that = {1, cos x, cos 2x, ..., cos mx, sin x, sin 2x, sin nx, ...} is an orthogonal set in a Hilbert space defined in C([0, 27]) and m, n are positive integers. (b) Demonstrate that a periodic function f(x) with a period of 27 can be expressed as a linear combination of ß by f(x)= a₁ + Σ(a, cos nx+b, sin nx) and find such a; and b¡. Such an + 2(a₂ n=1 (c) ... expression is called as Fourier series after the name of Joseph Fourier, a French mathematician. Prove that if f(x) is an even function, then all b;'s are 0.
A periodic function is defined by f(x + p) = f(x), where p is the period of f(x). Assume a periodic function is with period 27 and the inner product of 2 periodic functions is defined as 1 2π (1₁8) = = = = √ ₁²" 1 f.gdt. 2π Jo (a) Please verify that = {1, cos x, cos 2x, ..., cos mx, sin x, sin 2x, sin nx, ...} is an orthogonal set in a Hilbert space defined in C([0, 27]) and m, n are positive integers. (b) Demonstrate that a periodic function f(x) with a period of 27 can be expressed as a linear combination of ß by f(x)= a₁ + Σ(a, cos nx+b, sin nx) and find such a; and b¡. Such an + 2(a₂ n=1 (c) ... expression is called as Fourier series after the name of Joseph Fourier, a French mathematician. Prove that if f(x) is an even function, then all b;'s are 0.
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
![2.
A periodic function is defined by f(x + p) = f(x), where p is the period of f(x). Assume a
periodic function is with period 27 and the inner product of 2 periodic functions is defined as
1
•2π
27 10²t* f.gdt.
(f.g) =
(a)
Please verify that = {1, cos x, cos 2x, ..., cos mx, .... sin x, sin 2x, ..., sin nx, ...} is
an orthogonal set in a Hilbert space defined in C([0, 27]) and m, n are positive integers.
(b)
Demonstrate that a periodic function f(x) with a period of 27 can be expressed as a linear
combination of ß by f(x) = a₁ + Σ(a, cos nx+b, sin nx) and find such a; and b¡. Such an
00
n=1
(c)
(d)
(e)
expression is called as Fourier series after the name of Joseph Fourier, a French
mathematician.
Prove that if f(x) is an even function, then all b;'s are 0.
What happens if f(x) is an odd function?
Express the following periodic function by a Fourier series.
f(x)
k
THE
0
-k
- [[]]
2π
(a) The given function f(x) (Periodic rectangular wave)
1
Using the result in (e), calculate the value of 1-
3
+
1 1 1
--+-
5 7 9](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56658202-f4b1-4176-ad55-5e6ba5c24317%2F2a4ddbe7-afa8-4e37-b7b1-ed21a69a338f%2F0anxkce_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2.
A periodic function is defined by f(x + p) = f(x), where p is the period of f(x). Assume a
periodic function is with period 27 and the inner product of 2 periodic functions is defined as
1
•2π
27 10²t* f.gdt.
(f.g) =
(a)
Please verify that = {1, cos x, cos 2x, ..., cos mx, .... sin x, sin 2x, ..., sin nx, ...} is
an orthogonal set in a Hilbert space defined in C([0, 27]) and m, n are positive integers.
(b)
Demonstrate that a periodic function f(x) with a period of 27 can be expressed as a linear
combination of ß by f(x) = a₁ + Σ(a, cos nx+b, sin nx) and find such a; and b¡. Such an
00
n=1
(c)
(d)
(e)
expression is called as Fourier series after the name of Joseph Fourier, a French
mathematician.
Prove that if f(x) is an even function, then all b;'s are 0.
What happens if f(x) is an odd function?
Express the following periodic function by a Fourier series.
f(x)
k
THE
0
-k
- [[]]
2π
(a) The given function f(x) (Periodic rectangular wave)
1
Using the result in (e), calculate the value of 1-
3
+
1 1 1
--+-
5 7 9
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 4 steps with 3 images
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,
