Using a split-step fourier method for a radial equation

Question

I have the common heat equation: $$\partial_tU=\nabla^2U$$ In one dimension I can solve it using a split-step fourier method: $$U(x, t+\delta t)=F^{-1}\left(\exp(-i\delta tk^2)F(U)\right)$$ Now I know that in my case $U$ is radially symmetric, and in two dimensions: $$\partial_tU=\left(\frac{1}{r}\partial_r+\partial^2_r\right)U$$ Is there a way to still use the equation above with a fourier transformation in one dimension, or do I have to convert the equation into a cartesian coordinate system $$\partial_tU=\left(\nabla^2_x+\nabla^2_y\right)U$$ and apply a fourier transformation in two dimensions to it?