# Rectangular summation of multiple Fourier series and multi-parametric capacity

## TL;DR

This paper studies the summability and divergence of multiple Fourier series in the Dirichlet space of the polydisc, characterizing exceptional sets via multi-parametric logarithmic capacity and extending results to vector-valued functions.

## Contribution

It establishes the summability of multiple Fourier series outside sets of zero capacity and constructs divergent series on such sets, also extending results to vector-valued functions.

## Key findings

- Fourier series are summable outside zero capacity sets.
- Constructs divergent Fourier series on zero capacity sets.
- Characterizes exceptional sets for Dirichlet space functions.

## Abstract

We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a set of zero multi-parametric logarithmic capacity. Conversely, given a compact set in the torus of zero capacity, we construct a Fourier series in the class which diverges on this set, in the sense of Pringsheim. We also prove that the multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variation and radial limits of Dirichlet space functions. As a by-product of the methods of proof, the results also hold in the vector-valued setting.