On the Norms of the Integral Means of Spherical Fourier Sums

The paper deals with the spherical Fourier sums $S_r(f,x)=\sum_{\|k\|\le r}\widehat f(k)e^{ik\cdot x}$ of a periodic function $f$ in $m$ variables and the strong integral means of these sums $((\int_0^R |S_r(f,x)|^p \,dr)/R)^{1/p}$ for $p\ge1$. We establish the exact growth order as $R\to+\infty$ of the corresponding operators, i.e., the growth order of the quantities $\sup_{|f|\le 1}((\int_0^R |S_r(f,0)|^p\, dr)/R)^{1/p}$. The upper and lower bounds differ by their coefficients, which depend only on the dimension $m$. A sufficient condition on the function ensuring the uniform strong $p$-summability of its Fourier series is given.