On Convergence of Multiple Fourier Series for Piecewise Smooth Radial Functions in -.PI.,.PI. n (n.GEQ.3).

Abstract

For a function f ∈L1 ([−π,π] n) we denote by (SRf ) (x) (R > 0) the spherical partial sums of multiple Fourier series of f defined by (SR f )(x)=∑|k|≤Rf^kei( k,x) and let f (x)=F(|x|) be radial with support in { |x| ≤ a } (0 < a ≤ π ). In this note, when n ≥ 3, we prove that, for F∈Cl +2([0,a]) and l =[(n−3)⁄2], a necessary and sufficient condition under which limR→∞(SRf )(0) exists is that F(a)=F'(a)=…=F(l )(a)=0.