  We provide estimates for weighted Fourier sums of integrable functions
defined on the sphere when the weights originate from a multiplier operator
acting on the space where the function belongs. That implies refined estimates
for weighted Fourier sums of integrable kernels on the sphere that satisfy an
abstract H\"{o}lder condition based on a parameterized family of multiplier
operators defining an approximate identity. This general estimation approach
includes an important class of multipliers operators, namely, that defined by
convolutions with zonal measures. The estimates are used to obtain decay rates
for the eigenvalues of positive integral operators on L^2(Sm) and generated by
a kernel satisfying the H\"{o}lder condition based on multiplier operators on
L^2(Sm).
