Convergence of Fourier Series - Multiple Dimensions

Multiple Dimensions

Upon examining the equivalent problem in more than one dimension, it is necessary to specify the precise order of summation one uses. For example, in two dimensions, one may define

which are known as "square partial sums". Replacing the sum above with

lead to "circular partial sums". The difference between these two definitions is quite notable. For example, the norm of the corresponding Dirichlet kernel for square partial sums is of the order of while for circular partial sums it is of the order of .

Many of the results true for one dimension are wrong or unknown in multiple dimensions. In particular, the equivalent of Carleson's theorem is still open for circular partial sums. Almost everywhere convergence of "square partial sums" (as well as more general polygonal partial sums) in multiple dimensions was established around 1970 by Charles Fefferman.