# Sum of a periodic function

• May 17th 2011, 08:39 AM
bkarpuz
Sum of a periodic function
Dear MHF members,

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

Thanks.
bkarpuz

Exercise 19 in Chapter 3 of W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Science/Engineering/Math, 3rd edition, 1976.
• May 17th 2011, 09:20 AM
Tinyboss
I assume f satisfies some continuity condition, else it's definitely not true. Hint: show that $\displaystyle \{x+n\alpha\text{ mod }2\pi\}_{n\in\mathbb{N}_+}$ is dense in [0,2pi].
• May 17th 2011, 11:27 AM
Drexel28
Quote:

Originally Posted by bkarpuz
Dear MHF members,

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

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