δ-Rescaled pure super greedy algorithm with respect to Riesz dictionary

We propose the [Formula: see text]-Rescaled Pure Super Greedy Algorithm ([Formula: see text]-RPSGA) for approximation with respect to a dictionary [Formula: see text] in a Hilbert space. We estimate the upper bound of the error for the [Formula: see text]-RPSGA when [Formula: see text] is a Riesz dictionary. Our results show that the convergence rate of the [Formula: see text]-RPSGA on the closure of the convex hull of [Formula: see text] is optimal. Then, we design the [Formula: see text]-Rescaled Pure Super Greedy Learning Algorithm ([Formula: see text]-RPSGLA) for the kernel-based regression in supervised learning. We obtain the convergence rates of the [Formula: see text]-RPSGLA for continuous kernels. When the kernel is infinitely smooth, we derive a convergence rate that can be arbitrarily close to the best rate [Formula: see text] under a mild assumption of the regression function.