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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1995 • Question 4

Question

Math 6210 • Real Analysis Fall 1995 • Question 4

Recall the \(k\)th Fourier coefficient of \(f\in L^1([-\pi ,\pi ])\) is \[ \hat f(k)=\frac {1}{2\pi }\int _{-\pi }^{\pi }f(t)e^{-ikt}\,dt. \]

(1) If \(f(t)=t\) for \(-\pi \le t<\pi \), show that \(\hat f(k)=(-1)^{k+1}\frac {1}{ik}\) for \(k\ne 0\) and \(\hat f(0)=0\);

(2) Show that \(\sum _{k=1}^{\infty }k^{-2}=\frac {\pi ^2}{6}\).

Raw LaTeX

Recall the \(k\)th Fourier coefficient of
\(f\in L^1([-\pi,\pi])\) is
\[
\hat f(k)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-ikt}\,dt.
\]

(1) If \(f(t)=t\) for \(-\pi\le t<\pi\), show that
\(\hat f(k)=(-1)^{k+1}\frac{1}{ik}\) for \(k\ne 0\) and \(\hat f(0)=0\);

(2) Show that \(\sum_{k=1}^{\infty}k^{-2}=\frac{\pi^2}{6}\).

Topic Tags

Fourier Analysis and Convolution Lebesgue Integration Lp Spaces and Sequence Spaces