When harmonic spherical splines are used to interpolate and predict discret
ely given data we are confronted with the problem of solving symmetric posi
tive definite systems involving as many equations as the number of data. Du
e to harmonicity, these systems are dense and thus iterative methods should
be preferred to direct ones for large data sets. Nevertheless iterative so
lvers may converge very slowly or fail to converge. This paper develops a c
lass of preconditioners based on sparse (banded) symmetric positive definit
e approximations to the Gram matrices of harmonic kernels, the sparse appro
ximations being defined as Gram matrices of locally supported approximation
s to the kernels. After recalling the basic framework of harmonic spherical
splines, truncated Legendre coefficients with closed-form expressions are
presented. Numerical results on the use of the sparse approximates as preco
nditioners for the conjugate gradient method are shown and demonstrate the
efficiency of these preconditioners.