  A linear operator in a Hilbert space defined through the form of Riesz
representation naturally introduces a reproducing kernel Hilbert space
structure over the range space. Such formulation, called H-HK formulation in
this paper, possesses a built-in mechanism to solve some basic type problems in
the formulation by using the basis method, that include identification of the
range space, the inversion problem, and the Moore-Penrose pseudo- (generalized)
inversion problem. After a quick survey of the existing theory, the aim of the
article is to establish connection between this formulation with sparse series
representation, and in particular with one called pre-orthogonal adaptive
Fourier decomposition (POAFD), the latter being one, most recent and well
developed, with great efficiency and wide and deep connections with traditional
analysis. Within the matching pursuit methodology the optimality of POAFD is
theoretically guaranteed. In practice POAFD offers fast converging numerical
solutions.
