(a) Please verify that = {1, cos x, cos 2x, ..., Cosmx, ..., 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 sin x, sin 2x, .... sin nx, ...} is (c) combination of ß by f(x)= a + Ž(ª„ cosnx + b² sinnx) and find such a; and b¡. Such an n=1 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) Please verify that = {1, cos x, cos 2x, ..., Cosmx, ..., 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 sin x, sin 2x, .... sin nx, ...} is (c) combination of ß by f(x)= a + Ž(ª„ cosnx + b² sinnx) and find such a; and b¡. Such an n=1 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
A,B, and C only
![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%2F3439b7b3-594d-4507-84ea-8dc97bf9f49d%2Feic2xp_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
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