  In this paper, we propose a unified theoretical and practical spherical
approximation framework for functional inverse problems on the hypersphere.
More specifically, we consider recovering spherical fields directly in the
continuous domain using functional penalised basis pursuit problems with gTV
regularisation terms. Our framework is compatible with various measurement
types as well as non-differentiable convex cost functionals. Via a novel
representer theorem, we characterise their solution sets in terms of spherical
splines with sparse innovations. We use this result to derive an approximate
canonical spline-based discretisation scheme, with vanishing approximation
error. To solve the resulting finite-dimensional optimisation problem, we
propose an efficient and provably convergent primal-dual splitting algorithm.
We illustrate the versatility of our framework on real-life examples from the
field of environmental sciences.
