9. Let f(æ)= X[a,b) (x) be the characteristic function of the interval [a, b] c [-7, 7], that is, Į 1 if r € (a, b), 0 otherwise. X[a,b} (x) = %3D (a Show that the Fourier series of ƒ is given by b - a f(r) ~ 27 e-ina – e-inb eina 2nin The sum extends over all positive and negative integers excluding 0.
9. Let f(æ)= X[a,b) (x) be the characteristic function of the interval [a, b] c [-7, 7], that is, Į 1 if r € (a, b), 0 otherwise. X[a,b} (x) = %3D (a Show that the Fourier series of ƒ is given by b - a f(r) ~ 27 e-ina – e-inb eina 2nin The sum extends over all positive and negative integers excluding 0.
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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![9. Let f(r) = x\a,b) (x) be the characteristic function of the interval [a, b] C
[- π, π , that is
{
S1 if r€ [a,b],
0 otherwise.
X[a,8)(x)
(a Show that the Fourier series of ƒ is given by
e-ina – e-inb
einz
b - a
f(r) ~
27
2тin
The sum extends over all positive and negative integers excluding 0.
(b) Show that if a + -r or b# n and a + b, then the Fourier series does not
converge absolutely for any r. [Hint: It suffices to prove that for many
values of n one has | sin n8o| 2c> 0 where 60 = (b – a)/2.]
(c) However, prove that the Fourier series converges at every point r. What
happens if a = -r and b= n?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F833481a2-df8c-4805-95a2-f24b64ba619f%2Fddc70410-35f4-4dc2-9cb6-03170cad69f1%2F1taiemf_processed.png&w=3840&q=75)
Transcribed Image Text:9. Let f(r) = x\a,b) (x) be the characteristic function of the interval [a, b] C
[- π, π , that is
{
S1 if r€ [a,b],
0 otherwise.
X[a,8)(x)
(a Show that the Fourier series of ƒ is given by
e-ina – e-inb
einz
b - a
f(r) ~
27
2тin
The sum extends over all positive and negative integers excluding 0.
(b) Show that if a + -r or b# n and a + b, then the Fourier series does not
converge absolutely for any r. [Hint: It suffices to prove that for many
values of n one has | sin n8o| 2c> 0 where 60 = (b – a)/2.]
(c) However, prove that the Fourier series converges at every point r. What
happens if a = -r and b= n?
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