I want to compute the following limit $$ \lim_{x\longrightarrow +\infty} \frac{1}{x}\int_0^x \sqrt{\mu+\left(2-\cos (2\pi y) - \cos(2\sqrt{2}\pi y)\right)^2}\;dy$$ where $\mu>0$ is a constant. I don't think about anyway to estimate that.
Edited: One way I think about it is try to use approximation by trigonometric polynomials, then problem will be reduced to the mean of the function, here $$ H(x,y) = \sqrt{\mu+\left(2-\cos (2\pi x) - \cos(2\pi y)\right)^2}$$ In one dimension we have any $f\in C(\mathbb{T})$ can be uniformly approximated by trigonometric polynomials. I am not sure we have that in $\mathbb{R}^n$ with $n\geq 2$.