Reproducing kernel Banach spaces with the ℓ1 norm

Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the following properties: (1) B possesses an ℓ1 norm in the sense that B is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous linear functionals on B and are representable through a bilinear form with a kernel function; and (3) regularized learning schemes on B satisfy the linear representer theorem. Examples of kernel functions admissible for the construction of such spaces are given.