# Thread: Sum of a periodic function

1. ## Sum of a periodic function

Dear MHF members,

Question. Suppose that $f:\mathbb{R}\to\mathbb{R}$ is a continuous periodic function with a period of $2\pi$. If $\alpha/\pi$ is irrational, then for any $x\in\mathbb{R}$ show that the sum $\frac{1}{N}\sum_{n=1}^{N}f(x+n\alpha)$ converges to $\frac{1}{2\pi}\int_{-\pi}^{\pi}f(u)\mathrm{d}u$.

Thanks.
bkarpuz

Exercise 19 in Chapter 3 of W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Science/Engineering/Math, 3rd edition, 1976.

2. I assume f satisfies some continuity condition, else it's definitely not true. Hint: show that $\{x+n\alpha\text{ mod }2\pi\}_{n\in\mathbb{N}_+}$ is dense in [0,2pi].

3. Originally Posted by bkarpuz
Dear MHF members,

Question. Suppose that $f:\mathbb{R}\to\mathbb{R}$ is a continuous periodic function with a period of $2\pi$. If $\alpha/\pi$ is irrational, then for any $x\in\mathbb{R}$ show that the sum $\frac{1}{N}\sum_{n=1}^{N}f(x+n\alpha)$ converges to $\frac{1}{2\pi}\int_{-\pi}^{\pi}f(u)\mathrm{d}u$.

Thanks.
bkarpuz

Exercise 19 in Chapter 3 of W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Science/Engineering/Math, 3rd edition, 1976.
Tinyboss gave you an idea how to prove this. This theorem is a specific case of one of the equidistribution theorems