On strong means of spherical Fourier sums

Abstract:

The spherical Fourier sums \[ S_n(f,x)=\sum _{\|k\|\leq n}\widehat f(k) e^{ik\cdot x} \] of a periodic function $f$ in $m$ variables and their strong means \[ H_{n,p}(f,x)=\bigg (\frac 1n\sum _{j=0}^{n-1}|S_j(f,x)|^p\bigg )^{\frac 1p} \quad \text {for}\quad p\geq 1 \] are considered. In contrast to the one-dimensional case treated by Hardy and Littlewood, for $m\geq 2$ the norms $\sup _{|f|\leq 1}H_{n,p}(f,0)$ are not bounded. The sharp order of growth of these norms is found (the upper and lower bounds differ by a factor depending only on the dimension $m$).