On Convergence of Multiple Fourier Series for Piecewise Smooth Radial Functions in [−π,π]ⁿ (n ≥ 3)

Abstract

For a function f ∈L¹ ([−π,π] ⁿ) we denote by (S_Rf ) (x) (R > 0) the spherical partial sums of multiple Fourier series of f defined by (S_R f )(x)=∑_|k|≤Rf^_ke^i( 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∈C^l +2([0,a]) and l =[(n−3)⁄2], a necessary and sufficient condition under which lim_R→∞(S_Rf )(0) exists is that F(a)=F'(a)=…=F^(l )(a)=0.