 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2011/d/d3/b/i/solution

H = \int_{0}^{π} ∣ h (t) ∣ d t < \infty

\int_{0}^{\infty} e^{- β x} ∣ h (x) ∣ d x = k = 0 \sum \infty e^{- β kπ} \int_{0}^{π} e^{- βt} ∣ h (t) ∣ d t \leq H k = 0 \sum \infty e^{- β kπ} = \frac{H}{1 - e ^{- β π}} < \infty.

Thus $f$ is Lebesgue integrable for every

β > 0

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