# Periodic function's integral

Let $$b > 0 \in \mathbb R$$, a continuous function $$f : [0, b] \to \mathbb R$$, and a periodic and continuous function $$p : \mathbb R \to \mathbb R$$ of period $$T = 1$$.

1. If $$p \geq 0$$ on $$\mathbb R$$ et $$\int_0^1 p(t) dt = 1$$, prove that $$\lim_{n \to \infty} \int_0^b f(t)p(nt) dt = \int_0^bf(t) dt.$$

I was told that we should use Riemann/Darboux sums, is there any another way to do it ? If not how to do it with Riemann sums ?

• Are you familiar with Lebsgue integration? Feb 22 at 20:20
• No sorry, only Riemann sums and other well known properties Feb 22 at 20:21

$$1)$$ Prove that the statement is true for $$p(x)=e^{2\pi i x}$$.
$$2)$$ Prove that it is true for trigonometric polynomials $$p(x)=\sum\limits_{j=-m}^ma_je^{2\pi i jx} \quad with \quad\sum\limits_{j=-m}^ma_j=1,$$ ie, for any element belonging to the span of $$\{e^{2\pi inx}:n\in\mathbb{Z}\}$$ with norm equal to 1.
$$3)$$ Use that $$\{e^{2\pi inx}:n\in\mathbb{Z}\}$$ is an orthonormal complete system, which implies that $$\{e^{2\pi inx}:n\in\mathbb{Z}\}$$ is dense in $$L^2(\mathbb{T})$$. In other words, any function $$p$$ fulfilling the hypothesis can be approximated by a sequence of trigonometric polynomials that satisfy the original property.
Note: I solved the problem assuming that $$f$$´s domain was $$[0,1]$$ instead of $$[0,b]$$, but this should be easy to fix.