Summary: In a Hilbert space a linear operator defined through an inner product kernel has a natural reproducing kernel structure. In the present paper we first define what we call the \(\mathcal{H}\)-\(H_K\) formulation that stands as an axiomatic basis of the study. There is a built-in mechanism in the \(\mathcal{H}\)-\(H_K\) formulation that can straightforwardly solve three basic type problems, namely, the image function identification, inverse problem, and Moore-Penrose pseudo-inverse problem. After a summary on the classical basis method we introduce the POAFD (pre-orthogonal adaptive Fourier decomposition) “non-basis” method in the \(\mathcal{H}\)-\(H_K\) formulation. We give historical notes, theoretical foundations, as well as algorithm principles of POAFD. The maximal selection principle of POAFD makes itself be the best among all the existing matching pursuit methods in the one-step-optimal-selection category. It, therefore, can be used to give fast converging sparse numerical solutions of approximation, ordinary and partial differential equations, and optimization problems.