Derivative reproducing properties for kernel methods in learning theory

The regularity of functions from reproducing kernel Hilbert spaces (RKHSs) is studied in the setting of learning theory. We provide a reproducing property for partial derivatives up to order s   when the Mercer kernel is C2sC2s. For such a kernel on a general domain we show that the RKHS can be embedded into the function space CsCs. These observations yield a representer theorem for regularized learning algorithms involving data for function values and gradients. Examples of Hermite learning and semi-supervised learning penalized by gradients on data are considered.