Slepian Basis for Efficient Representation of Order-Limited Signals on the Sphere

The Slepian concentration problem is a classical problem in signal processing concerned with finding basis functions that are simultaneously concentrated in spatial and spectral domains. Unlike spherical harmonic basis functions, Slepian functions enable localized analysis and have been widely used for representing data restricted to regions on the sphere. However, existing Slepian representations retain all orders of the basis, which leads to redundancy, particularly when the region of interest exhibits plausible azimuthal symmetry. In this work, we propose an order-limited Slepian basis that systematically excludes redundant higher-order components while preserving accuracy. We provide an analytical formulation of this order-limited basis, establish its properties, and derive expressions for two special cases: polar cap regions and limited colatitude–longitude regions. We further establish key mathematical properties of the proposed order-limited basis, including orthogonality, completeness, and concentration characteristics within the defined subspace. Using Earth’s topographic map and climate data as applications, we demonstrate that the proposed formulation achieves efficient signal representation with substantially fewer basis functions compared to conventional Slepian approaches. This highlights order-limited Slepian functions as an efficient alternative for localized signal analysis on the sphere.