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Riemann-Lebesgue lemma

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Recall that S_N (f)(x)= sum (n=-N to N) c_n e^{inx}=
1/2pi integral (from -pi to pi) f(x-t)sin ((N+1/2)t)/sin(t/2) dt

Prove that if f in R[-pi, pi] and integral (from -1 to 1) |f(t)/t)|dt < infinity (convergent) then
lim( as N goes to infinity) S_N(f)(0)=0

Hint: Use the Riemann-Lebesgue lemma.

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Solution Summary

Fourier series question is evaluated in this solution.