Signal analysis on the ball: Design of optimal basis functions with maximal multiplicative concentration in spatial and spectral domains

Abstract

In this work, we design a set of complete orthonormal optimal basis functions for signals defined on the ball. We design the basis functions by maximizing the product of their energy concentration in some spatial region and that in some spectral region. The optimal basis functions are designed as a linear combination of space-limited functions with maximal concentration in the spectral region and band-limited functions with maximal concentration in the spatial region. The proposed optimal basis functions are shown to form a complete set for signal representation in a subspace formed by the vector sum of the subspaces spanned by space-limited and band-limited functions. We also formulate an integral operator which projects the signal to the joint subspace and maximizes the product of energy concentrations in harmonic as well as spatial domains. Furthermore, with the help of some properties of proposed optimal basis functions we show that these functions are the only eigenfunction of the integral operator.