Rectangular Summability of Higher Dimensional Fourier Series

In this chapter, we investigate the rectangular summability of d-dimensional Fourier series. We consider two types of convergence, the so-called restricted and unrestricted convergence. In the first case, $$n \in {\mathbb N}^{d}$$ n ∈ N d is in a cone or a cone-like set and $$n\rightarrow \infty $$ n → ∞ while in the second case, we have $$n \in {\mathbb N}^{d}$$ n ∈ N d and $$\min (n_1,\ldots ,n_d)\rightarrow \infty $$ min ( n 1 , … , n d ) → ∞ , which is called Pringsheim’s convergence. Similarly, we consider two types of maximal operators, the restricted one defined on a cone or cone-like set and the unrestricted one defined on $${\mathbb N}^{d}$$ N d . We prove similar results as for the $$\ell _q$$ ℓ q -summability. In the restricted case, we use the Hardy space $$H_p^\Box ({\mathbb T}^d)$$ H p □ ( T d ) and in the unrestricted case a new Hardy space $$H_p({\mathbb T}^d)$$ H p ( T d ) .