  The present work develops certain analytical tools required to construct and
compute invariant kernels on the space of complex covariance matrices. The main
result is the $\mathrm{L}^1\,-$Godement theorem, which states that any
invariant kernel, which is (in a certain natural sense) also integrable, can be
computed by taking the inverse spherical transform of a positive function.
General expressions for inverse spherical transforms are then provided, which
can be used to explore new families of invariant kernels, at a rather moderate
computational cost. A further, alternative approach for constructing new
invariant kernels is also introduced, based on Ramanujan's master theorem for
symmetric cones.
