4. i) Prove that the series sin nx n2 n=1 converges pointwise on R, and uniformly on any finite interval.. Call its sum f(x). ii) Obtain the Fourier series of the function g defined by g(x) = | f(t)dt, justifying your method by referring to appropriate theorems. iii) What can you say about continuity and smoothness of the function g? Explain.
4. i) Prove that the series sin nx n2 n=1 converges pointwise on R, and uniformly on any finite interval.. Call its sum f(x). ii) Obtain the Fourier series of the function g defined by g(x) = | f(t)dt, justifying your method by referring to appropriate theorems. iii) What can you say about continuity and smoothness of the function g? Explain.
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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Transcribed Image Text:4.
i) Prove that the series
Σ
sin nx
n2
n=1
converges pointwise on R, and uniformly on any finite interval.. Call
its sum f(x).
ii) Obtain the Fourier series of the function g defined by
g(z) = | f(t)dt,
justifying your method by referring to appropriate theorems.
iii) What can you say about continuity and smoothness of the function
g? Explain.
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