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  3. this question is from p59 2 of quotfourier analysisquot by...

Question: this question is from p59 2 of quotfourier analysisquot by...

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This question is from p.59 #2 of "Fourier Analysis" by Stein-Shakarchi.

show how the symmetries of a function imply certain ise we its Fourier coefficients. Let f be a 2m-periodic Riemann integrable that the Fourier series of the function f can be written as f(9) ~ f(0) + Σ[f(n) + f(-n)] cosne + i[f(n)-/(-n)] sin no. that if f is even, then f(n) -(-n), and we get a cosine series. se that f(e+π)-/(0) for all θ E R. Show that f(n) = 0 for all 2. In this exercise In des of . function defined on R n21 ve that if f is odd, then f(n)m), and we get a sine series. (d) Suppose odd n. e) Show that f is real-valued if and only if f(n) = f(-n) for all n.

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