Single-scale f-greedy convergence outside the native space

Establish a convergence result for the single-scale f-greedy algorithm when the target function lies outside the native space of the kernel, thereby enabling an analysis of the multiscale f-greedy algorithm for such targets.

Background

The theoretical analysis of the multiscale f-greedy algorithm assumes that the target function belongs to the kernel’s native space. The paper identifies targets outside that space as an important regime, noting that the numerical experiment involving a kink suggests that the algorithms may still perform well there. Extending the multiscale theory requires a corresponding unresolved convergence result for the single-scale f-greedy algorithm.