Suppose that f is Riemann integrable on $\displaystyle (-\pi,\pi )$. Prove that:
$\displaystyle \lim_{n \to \infty} \int_{-\pi}^{\pi} \! f(x)cos(nx)dx=0$.
I can't see how to start this question, could someone help?
Thanks.
Suppose that f is Riemann integrable on $\displaystyle (-\pi,\pi )$. Prove that:
$\displaystyle \lim_{n \to \infty} \int_{-\pi}^{\pi} \! f(x)cos(nx)dx=0$.
I can't see how to start this question, could someone help?
Thanks.
This result is known as the Riemann–Lebesgue lemma. If the function f is differentiable, then Also sprach Zarathustra's hint provides a quick and easy proof. If f is continuous then the R–L lemma is an easy consequence of Fejér's theorem. But if all you know about f is that it is Riemann integrable, then you have to follow girdav's advice and approximate f by a simple function.
The R–L lemma is easy to prove for the characteristic function of an interval (by explicitly calculating the integral). The result then follows for a simple function (which is just a finite linear combination of characteristic functions). Finally, to approximate a Riemann integrable function by a simple function, take a Riemann sum corresponding to a suitably fine dissection of the interval.