Signal concentration on unit sphere: A local k-th moment zenithal energy concentration measure

We define a k-th moment zenithal energy concentration measure for a polar cap region on the unit sphere. The optimal functions are obtained with the maximal local k-th moment zenithal measure (LMZM). Compared to the spherical Slepian function in the cap region and the optimal function with the minimal global k-th moment zenithal measure (GMZM), we show that: 1) as k increases, the tail of the function with LMZM decays more quickly; 2) the main lobe of the optimal function with LMZM moves as the maximum colatitude of the cap increases; while the spherical Slepian function does not change much and the function with GMZM is by definition invariant; and 3) LMZM has no obvious advantage over the spherical Slepian function and the GMZM optimal function in both the energy concentration and tail decay rate.