Revisiting Slepian concentration problem on the sphere for azimuthally non-symmetric regions

The problems of filtering, spectral analysis and spectral estimation have been investigated on the sphere using azimuthally symmetric functions as kernels which treat all the directions uniformly. In this work, we extend the concentration problem on the sphere for an azimuthally non-symmetric spatial region on the sphere. Our approach is different in a sense that we obtain the family of spatially concentrated bandlimited mutually orthogonal functions by maximizing the contribution of spherical harmonics components of all degrees and orders within the spectral bandwidth. We also provide analysis of the eigenfunctions for different bandwidths and non-symmetric regions and illustrate the concentration of eigenfunctions with the help of examples. Also we formulate the definition of filtering using azimuthally non-symmetric functions. The proposed eigenfunctions can be used to revisit the problems of estimation, localized spectral analysis, smoothing and filter design on the sphere.