What question did this study set out to answer?

The aim is to establish conditions for the convergence of spherical partial sums of the Fourier inverse transform for radial functions.

April 1, 2026Open Access

On Convergence of Fourier Inverse Transforms for Piecewise Smooth Radial Functions in Rⁿ

Key Points

The aim is to establish conditions for the convergence of spherical partial sums of the Fourier inverse transform for radial functions.
Analyzed the behavior of spherical partial sums of the Fourier inverse transform for piecewise smooth radial functions.
Considered radial functions defined in terms of their smoothness and support.
Provided a detailed proof for necessary and sufficient conditions for convergence as R approaches infinity.
Identified conditions under which the spherical partial sums converge to zero.
Demonstrated that the smoothness of the function at the boundary influences convergence.
Concluded that if the function's derivatives vanish at the boundary, the spherical partial sums will tend to zero.

Abstract

For a function f∈Lᵖ (Rⁿ) (1≦p≦2), we denote by (SRf) (x) (R>0) the spherical partial sums of Fourier inverse transform of f defined by numerical formulaand 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^ (0, α), l= (n-3) /2, vanishing in a neighborhood of the origin, a necessary and sufficient condition under which we have lim_ (SRf) (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 (SRf) (x) for piecewise smooth radial function f.

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