In this paper, we provide a Banach-space formulation of supervised learning with generalized total-variation (gTV) regularization. We identify the class of kernel functions that are admissible in this framework. Then, we propose a variation of supervised learning in a continuous-domain hybrid search space with gTV regularization. We show that the solution admits a multi-kernel expansion with adaptive positions. In this representation, the number of active kernels is upper-bounded by the number of data points while the gTV regularization imposes an $\ell_1$ penalty on the kernel coefficients. Finally, we illustrate numerically the outcome of our theory.