Rectangular Summation of Multiple Fourier Series and Multi-parametric Capacity

Karl-Mikael Perfekt

doi:10.1007/s11118-020-09861-5

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.

Highlights

This article will consider unrestricted rectangular summation and other multi-parameter summation methods of the multiple Fourier series f (θ) ∼aα ei(α1θ1+···αnθn). (1) α∈ZnTo clarify this objective, note that there are several natural ways to form the partial sums of a multiple Fourier series
Note that there are several natural ways to form the partial sums of a multiple Fourier series
One can attempt to sum the series via square partial sums, spherical partial sums, lim aα ei(α1θ1+···αnθn), M →∞

Summary

Introduction

This article will consider unrestricted rectangular summation and other multi-parameter summation methods of the multiple Fourier series f (θ) ∼. For a series f (θ ) ∼ k∈Z akeikθ such that k∈Z |k||ak|2 < ∞, Beurling [8] showed that f (θ ) is summable for every θ ∈ T \ E, where E is a set of zero logarithmic capacity This was given a one-parameter generalization to multiple Fourier series by Lippman and Shapiro [17]. They proved that if f ∈ L1(Tn), n ≥ 2, is as in Eq 1 and satisfies that α∈Zn (α12 + · · · + αn2)|aα|2 < ∞, f (θ ) is summable with respect to spherical partial sums, except for on a set E ⊂ Tn of zero ordinary capacity (logarithmic capacity for n = 2 and Newtonian capacity for n ≥ 3, under the identification Tn (R/Z)n).

Multi-parametric Capacity

Convergence Theorems

Sharpness of results