On the L 2 -norm of periodizations of functions

We prove that the L2([0, 1]d × SO(d))-norm of periodizations of a function from L1(Rd) is equivalent to the L2(Rd)-norm of the function itself in higher dimensions. We generalize the statement for functions from Lp(Rd) where 1 ≤ p < 2d d+2 in the spirit of the Stein-Tomas theorem. 0. Introduction. Let f be a function from L(R). Define a family of its periodizations with respect to a rotated integer lattice: gρ(x) = ∑ ν∈Zd f(ρ(x− ν)) (1) for all rotations ρ ∈ SO(d). The main object of our study isG, the L([0, 1]d× SO(d))-norm of the family of periodizations, G = ∫