  We consider spherical Riesz means of multiple Fourier series and some
generalizations. While almost everywhere convergence of Riesz means at the
critical index $(d-1)/2$ may fail for functions in the Hardy space $h^1(\mathbb
T^d)$, we prove sharp positive results for strong summability almost
everywhere. For functions in $L^p(\mathbb T^d)$, $1<p<2$, we consider Riesz
means at the critical index $d(1/p-1/2)-1/2$ and prove an almost sharp theorem
on strong summability. The results follow via transference from corresponding
results for Fourier integrals. We include an endpoint bound on maximal
operators associated with generalized Riesz means on Hardy spaces $H^p(\mathbb
R^d)$ for $0<p<1$.

Classical Analysis and ODEs