Effective representation of the diffusion signal’s dependence on diffusion time is a sought-after, yet still unsolved challenge in diffusion MRI (dMRI). We propose a functional basis approach that is specifically designed to represent the dMRI signal in this four-dimensional space – varying over gradient strength, direction and diffusion time – that we call the multi-spherical space. In particular, we provide regularization tools imposing signal sparsity and signal smoothness to drastically reduce the number of measurements we need to probe the properties of this multi-spherical space.