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In this paper, we introduce one family of vectorial prolate spheroidal wavefunctions of real order $\alpha>-1$ on the unit ball in $R^3$, which satisfythe divergence free constraint, thus are termed as divergence free vectorialball PSWFs. They are vectorial eigenfunctions of an integral operator relatedto the finite Fourier transform, and solve the divergence free constrainedmaximum concentration problem in three dimensions, i.e., to what extent can thetotal energy of a band-limited divergence free vectorial function beconcentrated on the unit ball? Interestingly, any optimally concentrateddivergence free vectorial functions, when represented in series in vectorspherical harmonics, shall be also concentrated in one of the three vectorialspherical harmonics modes. Moreover, divergence free ball PSWFs are exactly thevectorial eigenfunctions of the second order Sturm-Liouville differentialoperator which defines the scalar ball PSWFs. Indeed, the divergence freevectorial ball PSWFs possess a simple and close relation with the scalar ballPSWFs such that they share the same merits. Simultaneously, it turns out thatthe divergence free ball PSWFs solve another second order Sturm-Liouville eigenequation defined through the curl operator $\nabla\times $ instead of thegradient operator $\nabla$.

Vectorial ball Prolate spheroidal wave functions with the divergence free constraint