Problem 5.10.1. Find the Fourier series of \(f(x)=x^{2}\) on \([-\pi ,\pi ]\), extended periodically. By evaluating at \(x=\pi \), deduce \(\sum n^{-2}=\pi ^{2}/6\); by evaluating at \(x=0\), deduce \(\sum (-1)^{n+1}n^{-2}=\pi ^{2}/12\). Where to start: \(x^{2}\) is even, so all \(b_n\) vanish and only the cosine integrals are needed.

Problem 5.10.2. Obtain the half-range sine and half-range cosine expansions of \(f(x)=x\) on \([0,\pi ]\), and sketch the function each represents on \([-2\pi ,2\pi ]\). Explain why one of them converges more slowly than the other. Where to start: the odd extension has a jump at \(x=\pm \pi \) and the even extension does not; compare the rate at which the coefficients decay.

Problem 5.10.3. Use Parseval’s identity on the series for \(f(x)=x\) to evaluate \(\sum ^{\infty }_{n=1}n^{-2}\), and on the series for the square wave to evaluate \(\sum ^{\infty }_{k=1}(2k-1)^{-2}\).

Problem 5.10.4. Show that the Fourier coefficients of a Riemann integrable function tend to zero, and explain how this is the Riemann–Lebesgue lemma of Corollary 5.4.2.

Problem 5.10.5. Show that if \(f\) is continuous and piecewise smooth with \(f(-\pi )=f(\pi )\) then its Fourier coefficients satisfy \(a_n,b_n = O(1/n^{2})\), whereas a jump discontinuity gives only \(O(1/n)\). Relate this to the Gibbs phenomenon of Figure 5.1. Where to start: integrate the coefficient formula by parts, and note that the boundary term vanishes only when the periodic extension is continuous.

Problem 5.10.6. Derive the complex form of the Fourier series from the real form, and verify that \(c_{-n}=\overline {c_n}\) when \(f\) is real-valued.

Problem 5.10.7. Compute the Fourier transform of the rectangular pulse \[f(t) = \begin {cases} 1, & |t|\leq a,\\ 0, & |t|>a,\end {cases}\] and show that \(\widehat {f}(\omega ) = 2\sin (a\omega )/\omega \). Comment on the reciprocal relationship between the width of the pulse and the width of its transform.

Problem 5.10.8. Prove the differentiation property \(\widehat {f'}(\omega )=i\omega \widehat {f}(\omega )\), stating the condition on \(f\) at \(\pm \infty \) that the proof requires. Where to start: integrate the defining integral by parts once.

Problem 5.10.9. Use the transform of the Gaussian to show that the convolution of two normal densities with variances \(\sigma _1^{2}\) and \(\sigma _2^{2}\) is normal with variance \(\sigma _1^{2}+\sigma _2^{2}\). Where to start: convolution becomes a product under the transform, and the product of two Gaussians in \(\omega \) is a Gaussian whose exponent adds.