  We show that the geodesic ray transform is injective on scalar functions on
spherically symmetric reversible Finsler manifolds where the Finsler norm
satisfies a Herglotz condition. We use angular Fourier series to reduce the
injectivity problem to the invertibility of generalized Abel transforms and by
Taylor expansions of geodesics we show that these Abel transforms are
injective. Our result has applications in linearized boundary rigidity problem
on Finsler manifolds and especially in linearized elastic travel time
tomography.
