Question 11.11: If f and |f| have improper integrals on R, prove that limn→∞...
If f and |f| have improper integrals on \mathbb{R}, prove that
\underset{n→∞}{\lim} \int_{−∞}^{∞} f (x) \cos nxdx = 0.
The function f(x) cos nx is clearly Lebesgue integrable on \mathbb{R} and we use Theorem 11.8 to assert that, for any ε > 0, there is a step function ψ = \Sigma_{i=1}^{k} c_{i}χ_{[a_{i} ,b_{i})}, which vanishes outside a compact interval, and which satisfies
\int_{\mathbb{R}} |f − ψ| dm < \frac{ε}{2} .
Thus
\left|\int_{−∞}^{∞}f (x) \cos nxdx\right| ≤ \left|\int_{−∞}^{∞}[f (x) − ψ(x)] \cos nxdx\right| + \left|\int_{−∞}^{∞}ψ(x) \cos nxdx\right|
< \frac{ε}{2} + \left|\sum\limits_{i=1}^{k}{c_{i}\int_{a_{i}}^{ b_{i}} \cos nxdx}\right|
= \frac{ε}{2} + \left|\sum\limits_{i=1}^{k} {\frac{c_{i}} {n}(\sin nb_{i} − \sin na_{i})}\right|
≤ \frac{ε}{2} + \frac{2}{n} \sum\limits_{i=1}^{k} {|c_{i}|} < ε for all n ≥ \frac{4}{ε}\sum\limits_{i=1}^{k} |c_{i}| .
Hence \lim_{n→∞} \int_{−∞}^{∞} f (x) \cos nxdx = 0. This result is sometimes referred to as the Riemann-Lebesgue lemma.