7.5E: An Application to Fourier Approximation Exercises

Exercises for 1

solutions

2

[ex:10_5_1] In each case, find the Fourier approximation \(f_{5}\) of the given function in \(\mathbf{C}[-\pi, \pi]\).

1. \(f(x) = \pi - x\)
2. \(f(x) = |x| = \left\{ \begin{array}{rl} x & \mbox{if } 0 \leq x \leq \pi \\ -x & \mbox{if } -\pi \leq x < 0 \end{array} \right.\)
3. \(f(x) = x^2\)
4. \(f(x) = \left\{ \begin{array}{rl} 0 & \mbox{if } -\pi \leq x < 0 \\ x & \mbox{if } 0 \leq x \leq \pi \end{array} \right.\)

2. \(\displaystyle \frac{\pi}{2} - \frac{4}{\pi} \left[ \cos x + \frac{\cos 3x}{3^2} + \frac{\cos 5x}{5^2} \right]\)
3. \(\displaystyle -\frac{2}{\pi} \left[ \cos x + \frac{\cos 3x}{3^2} + \frac{\cos 5x}{5^2} \right]\)

1. Find \(f_{5}\) for the even function \(f\) on \([-\pi, \pi]\) satisfying \(f(x) = x\) for \(0 \leq x \leq \pi\).
2. [Hint: If \(k > 1\), \(\int \sin x \cos(kx) \\ = \frac{1}{2} \left[ \frac{\cos[(k - 1)x]}{k - 1} - \frac{\cos[(k + 1)x]}{k + 1} \right]\).]

2. \(\displaystyle \frac{2}{\pi} - \frac{8}{\pi} \left[ \frac{\cos 2x}{2^2 - 1} + \frac{\cos 4x}{4^2 - 1} + \frac{\cos 6x}{6^2 - 1} \right]\)

1. Prove that \(\int_{-\pi}^{\pi} f(x)dx = 0\) if \(f\) is odd and that \(\int_{-\pi}^{\pi} f(x)dx = 2 \int_{0}^{\pi} f(x)dx\) if \(f\) is even.
2. Prove that \(\frac{1}{2}[f(x) + f(-x)]\) is even and that \(\frac{1}{2} [f(x) - f(-x)]\) is odd for any function \(f\). Note that they sum to \(f(x)\).

Show that \(\{1, \cos x, \cos(2x), \cos(3x), \dots\}\) is an orthogonal set in \(\mathbf{C}[0, \pi]\) with respect to the inner product \(\langle f, g \rangle = \int_{0}^{\pi} f(x)g(x)dx\).

\(\int \cos\ kx\ \cos\ lx\ dx\)
\(= \frac{1}{2} \left[ \frac{\sin[(k + l)x]}{k + l} - \frac{\sin[(k - l)x]}{k - l} \right]_0^{\pi} = 0\) provided that \(k \neq l\).

1. Show that \(\frac{\pi^2}{8} = 1 + \frac{1}{3^2} + \frac{1}{5^2} + \cdots\) using Exercise [ex:10_5_1](b).
2. Show that \(\frac{\pi^2}{12} = 1 - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \cdots\) using Exercise [ex:10_5_1](c).