The author obtains some pointwise convergence results for the spherical summation operators \[ S_R f(x) := \int_{| \xi| \leq R} \hat f(\xi) e^{2\pi i x \cdot \xi}d\xi \] in higher dimensions \(\mathbb R^n\), \(n > 1\). The pointwise convergence of this operator for \(L^2\) functions \(f\) is still an important open problem. However, the author shows that information on the set of divergence can be obtained if more regularity is assumed on the function. For instance, if \(f\) lies in the Sobolev space \(W^{\alpha,p}\) for \(2 \leq p < 2n/(n-1)\) and \(0 < \alpha \leq n/p\) then one has pointwise convergence outside of a set of \(C_{\alpha-\varepsilon,2}\)-capacity zero for all \(\varepsilon > 0\). (This is the case if one defines \(f\) properly, as the fractional integral of its \(\alpha^{th}\) derivative). The author conjectures that this \(\varepsilon\) can be removed, and that the convergence is uniform on open sets outside of the set of divergence; the former conjecture is proven when \(\alpha > (n-1)/2\) or on any open set where \(f\) is smooth. Once \(\alpha\) is greater than or equal to \((n-1)/2\) one has a form of the Riemann localization theorem, in that one has pointwise convergence in every open set where \(f\) is smooth, but this fails for \(\alpha\) less than \((n-1)/2\). Some further improvements are obtained assuming that the set of divergence is closed. The case \(p<2\) is also discussed; the situation there is worse. For instance, the author shows that if \(1 \leq p < 2-1/n\) and \(0 \leq \alpha < ((2-p)n-1)/2p\), then the partial sums can diverge everywhere.