  Targeting at sparse multi-task learning, we consider regularization models
with an $\ell^1$ penalty on the coefficients of kernel functions. In order to
provide a kernel method for this model, we construct a class of vector-valued
reproducing kernel Banach spaces with the $\ell^1$ norm. The notion of
multi-task admissible kernels is proposed so that the constructed spaces could
have desirable properties including the crucial linear representer theorem.
Such kernels are related to bounded Lebesgue constants of a kernel
interpolation question. We study the Lebesgue constant of multi-task kernels
and provide examples of admissible kernels. Furthermore, we present numerical
experiments for both synthetic data and real-world benchmark data to
demonstrate the advantages of the proposed construction and regularization
models.
