Fourier Expansion in d-dimensions

While reading, I came across an expansion of suitable functions as a Fourier series ~ and questions of convergence. I'm familiar with the argument for pointwise convergence of the symmetric sums defined by when f is, say differentiable, on , by convolving the function with the Dirichlet kernel. I decided to try and generalize the argument for the very special case of and 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 . So summing over symmetric cubes about the origin.

However, there are of course many norms to pick here. In fact, 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 , f is nice enough to make the question make sense, and , what affect does p have on , either in terms of utility of the expansion or in convergence in general?