9. Let f(x) = X[a.bj (x) be the characteristic function of the interval [a, b] C [-n, 7], that is, 1 ifr € [a,b], 0 otherwise. X[a,8](z) = . (a) Show that the Fourier series of f is given by b - a e-ina – e-inb f(z) ~ Σ 27 2лin n#0 The sum extends over all positive and negative integers excluding 0. (b) Show that if a± -x or b+ a 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| 2 c > 0 where 0o = (b – a)/2.]
9. Let f(x) = X[a.bj (x) be the characteristic function of the interval [a, b] C [-n, 7], that is, 1 ifr € [a,b], 0 otherwise. X[a,8](z) = . (a) Show that the Fourier series of f is given by b - a e-ina – e-inb f(z) ~ Σ 27 2лin n#0 The sum extends over all positive and negative integers excluding 0. (b) Show that if a± -x or b+ a 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| 2 c > 0 where 0o = (b – a)/2.]
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
[-7, 7], that is,
X[a.6) (2x) = { ! if x € [a, b],
0 otherwise.
(a) Show that the Fourier series of ƒ is given by
e-ina – e-inb
„inz
b - a
f(r) ~
27
2тin
n#0
The sum extends over all positive and negative integers excluding 0.
(b) Show that if a + –n or b# a 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 n6o| 2c> 0 where 09 = (b – a)/2.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F833481a2-df8c-4805-95a2-f24b64ba619f%2Fd4f35314-9836-4980-8efe-2c3d2589636a%2Fmn1yvko_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
[-7, 7], that is,
X[a.6) (2x) = { ! if x € [a, b],
0 otherwise.
(a) Show that the Fourier series of ƒ is given by
e-ina – e-inb
„inz
b - a
f(r) ~
27
2тin
n#0
The sum extends over all positive and negative integers excluding 0.
(b) Show that if a + –n or b# a 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 n6o| 2c> 0 where 09 = (b – a)/2.]
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