(Fourier series) (a) Show that the pointwise convergent series 00 sin(nx) n1/2 11=1 cannot converge uniformly to a square integrable function f in [-T, T). (b) Let f(x) be 27t periodic and piecewise smooth. Prove that its Fourier series converges uniformly and absolutely to f.

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Chapter2: Second-order Linear Odes
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Please solve both parts as soon as possible

(Fourier series)
(a)
Show that the pointwise convergent series
sin(nx)
n1/2
1n=1
cannot converge uniformly to a square integrable function f in [-T, T).
(b)
Let f(x) be 2n periodic and piecewise smooth. Prove that its Fourier series converges
uniformly and absolutely to f.
Transcribed Image Text:(Fourier series) (a) Show that the pointwise convergent series sin(nx) n1/2 1n=1 cannot converge uniformly to a square integrable function f in [-T, T). (b) Let f(x) be 2n periodic and piecewise smooth. Prove that its Fourier series converges uniformly and absolutely to f.
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