Approximation numbers of integral operators on the sphere

This work derives sharp estimates for approximation numbers of positive integral operators on the sphere when the generating kernel satisfies an abstract Hölder condition defined by spherical convolutions with uniformly bounded bi-zonal kernels. The estimates are obtained via finite rank operators defined by both, certain generalized Jackson kernels and the operators appearing in the Hölder condition. In the case the generating kernel of the integral operator is continuous, the estimates imply decay rates for the eigenvalues of the integral operator. Estimates for approximation numbers of integral operators acting on a Hilbert space has been a topic of study for a long time. The spectral theory surrounding the well-known Mercer’s theorem forces a basic decay to hold and better ones can be reached at the cost of additional assumptions on the generating kernel. Usually, a differentiability-like assumption on the kernel improves the basic decay quite a bit ([1]). The use of Hölder-type assumptions is not either uncommon or exclusive of the spherical setting, as one can verify in many research works (see for example, [2, 3]).