  This paper studies transfer learning for estimating the mean of random
functions based on discretely sampled data, where, in addition to observations
from the target distribution, auxiliary samples from similar but distinct
source distributions are available. The paper considers both common and
independent designs and establishes the minimax rates of convergence for both
designs. The results reveal an interesting phase transition phenomenon under
the two designs and demonstrate the benefits of utilizing the source samples in
the low sampling frequency regime. For practical applications, this paper
proposes novel data-driven adaptive algorithms that attain the optimal rates of
convergence within a logarithmic factor simultaneously over a large collection
of parameter spaces. The theoretical findings are complemented by a simulation
study that further supports the effectiveness of the proposed algorithms
