On Convergence of Fourier Inverse Transforms for Piecewise Smooth Radial Functions in R^n

概要

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For a function f∈L^p(R^n)(1≦p≦2), we denote by (S_Rf)(x)(R>0) the spherical partial sums of Fourier inverse transform of f defined by [numerical formula]and let =f(x)=F(|x|) be radial with support in {|x|≦α} (α>0). In this note, in particular, when n≧3, we give a detailed proof of the fact that, for smooth F∈C^<l+2>([0,α]), l=[(n-3)/2], vanishing in a neighborhood of the origin, a necessary and sufficient condition under which we have lim_<R→∞>(S_Rf)(0)=0 is the validity of F^(k)(α)=0 for all k=0, 1,...,l. This fact gives a negative answer to the localization problem concerning of (S_Rf)(x) for piecewise smooth radial function f.
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