The quantization problem is the problem of associating a non-commutative quantum algebra to a classical Poisson algebra in such a way that the commutator is related to the Poisson bracket. In a formal setting, this problem and its equivariant counterpart are well-understood, and equivariant formal deformation quantizations of semisimple coadjoint orbits were constructed by Alekseev--Lachowska from the Shapovalov pairing. Using the example of the 2-sphere, I will illustrate how their results can be used to obtain a family of equivariant non-formal products for a certain class of analytic functions on semisimple coadjoint orbits. This provides examples of quantizations in a Fréchet-algebraic setting.