## Fourier Expansion in d-dimensions

While reading, I came across an expansion of suitable functions $f:\mathbb{R}^d \to \mathbb{R}$ as a Fourier series $f(x)$ ~ $\displaystyle \sum_{k \in \mathbb{Z}^d} \hat{f}(k) e^{ik \cdot x}$ and questions of convergence. I'm familiar with the argument for pointwise convergence of the symmetric sums defined by $s_N(f; x) = \sum_{|k| \le N} \hat{f}(k) e^{ikx}$ when f is, say differentiable, on $\mathbb{R}$, by convolving the function with the Dirichlet kernel. I decided to try and generalize the argument for the very special case of $f \in C^{\infty}(\mathbb{R}^d)$ and $2 \pi$ periodic in each variable. However, my question then became the appropriate way to generalize the partial summations (or summations over d-tuples in general). The most natural choice for extending the argument seemed to be by replacing the absolute value with $|x|_{\infty} := \sup_{j=1,...,d} |x_j|$. So summing over symmetric cubes about the origin.

However, there are of course many norms to pick here. In fact, $|x|_p := (\sum_{j=1}^d |x_j|^p )^{\frac{1}{p}$ for all 1 <= p < \infty all provide viable summation candidates. While for my purposes the limiting case of p = \infty was sufficient, I'm curious about the differences for the intermediate cases. In particular, if $1 \le p < \infty$, f is nice enough to make the question make sense, and $s_N^p(f; x) := \sum_{|k|_p \le N} \hat{f}(k) e^{i k \cdot x}$, what affect does p have on $\lim_{N \to \infty} s^p_N(f;x)$, either in terms of utility of the expansion or in convergence in general?