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.