  We study transfer learning for estimating piecewise-constant signals when
source data, which may be relevant but disparate, are available in addition to
the target data. We first investigate transfer learning estimators that
respectively employ $\ell_1$- and $\ell_0$-penalties for unisource data
scenarios and then generalise these estimators to accommodate multisources. To
further reduce estimation errors, especially when some sources significantly
differ from the target, we introduce an informative source selection algorithm.
We then examine these estimators with multisource selection and establish their
minimax optimality. Unlike the common narrative in the transfer learning
literature that the performance is enhanced through large source sample sizes,
our approaches leverage higher observation frequencies and accommodate diverse
frequencies across multiple sources. Our theoretical findings are supported by
extensive numerical experiments, with the code available online, see
https://github.com/chrisfanwang/transferlearning
