On $\mathbb{T}$, we have approximation series $\{f_n\}$ of $f\in L^p,p\in[1,\infty)$ in the form of $f_n(x)=\sum_{|k|\le n}a_{n,k}e^{ikx}$, converging to $f$ in $L^p$. (for example, we can consider convolution with Fejer kernel)
Can we do something similar for $\mathbb{T}^n$ as well?
Summability is no problem, for example if $K_n(t)$ is the Fejer kernel then $K_n(s)K_n(t)$ is an approximate identity on $\Bbb T^2$. Now, convergence for multiple Fourier series is a serious problem...