This paper deals with Fourier series on classical groups. The spherical summability including spherical Riesz means at the critical index is considered. Let \(U_ n\) denote the unitary group of order n. In the case \(n=1\) Hardy and Littlewood established the following theorems. Theorem A. Let \(u\in L(U_ 1)\) and \(x_ 0\in {\mathbb{R}}\). If \[ (1)\quad \lim_{t\to 0^+}(1/t)\int^{t}_{-t}[u(x_ 0+\tau)-u(x_ 0)] d\tau =0 \] and (2) \(| a_ k| \leq C/| k|\) (k\(\neq 0)\) where \(a_ k\) (k\(\in {\mathbb{Z}})\) are Fourier coefficients of u and C is a constant independent of k, then the Fourier series of u converges to u at the point \(x_ 0.\)
Theorem B. If the conditions (1) and (2) in Theorem A are replaced respectively by \(\lim_{t\to 0}(u(x_ 0+t)-u(x_ 0))/\log t=0\) and \(| a_ k| \leq C/| k|^{\alpha}\) (k\(\neq 0)\) where \(\alpha >0\) is independent of \(k\in {\mathbb{Z}}\), then the conclusion still holds.
The authors generalize these theorems to the general case of classical groups, stating the proofs only on \(U_ n\). The Riemann principle of localization is also generalized for Riesz spherical means.