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Question 10.4.2: Suppose that f(x) = {1 − x, 0 < x ≤ 1, 0, 1 < x ≤ 2. (......

Suppose that

f(x)=\begin{cases}1 − x, & 0\lt x \leq 1, \\0, & 1\lt x \leq 2.\end{cases} (13)

As indicated previously, we can represent f either by a cosine series or by a sine series. Sketch the graph of three periods of the sum of each of these series for −6 ≤ x ≤ 6.

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In this example, L = 2, so the cosine series for f converges to the even periodic extension of f of period 4, whose graph is sketched in Figure 10.4.4.

Similarly, the sine series for f converges to the odd periodic extension of f of period 4. The graph of this function is shown in Figure 10.4.5.

FIGURE 10.4.4
FIGURE 10.4.5
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