Selecting the optimal Parameters Results in Double Interpolation: Double AFD

Let $f$ belong to the Hardy space $H^2(\mathbb{D})$ of the unit disc, and $e_a$ the normalized Szeg\"o (reproducing) kernel of $H^2(\mathbb{D}).$ It is well known that, due to the reproducing kernel property, for any distinct $n$ points $a_1,\cdots,a_n$ in $\mathbb{D}$ the orthogonal projection of $f$ into ${\rm span}\{e_{a_1},\cdots,e_{a_n}\},$ denoted as $P_{{\rm span}\{e_{a_1},\cdots,e_{a_n}\}}(f),$ interpolates $f$ at the points $a_k$'s. The present study further proves that if the $a_k$'s are optimally selected according to certain energy matching pursuit principle, then $P_{{\rm span}\{e_{a_1},\cdots,e_{a_n}\}}(f)$ double interpolates $f$ at the points $a_k$'s, or order $m=2$ interpolation, that is, $$P_{{\rm span}\{e_{a_1},\cdots,e_{a_n}\}}(f)(a_k)=f(a_k), \quad {\rm and}\quad P_{{\rm span}\{e_{a_1},\cdots,e_{a_n}\}}'(f)(a_k)=f'(a_k),\quad k=1,\cdots,n.$$ With the accordingly newly defined double Takenaka-Malmquist system, the norm convergence for $n\to \infty,$ the $n$-best approximation for $n$ being fixed, and the related boundary function interpolation are studied. The such generated new sparse representation, named as double AFD, is shown to outperform the classical AFD. Pointwise interpolations for orders $m>2,$ meaning to simultaneously interpolates all functions $f,f',\cdots,f^{(m-1)}$ at a set of $a_k$'s are, additionally, discussed. For the Hardy space of the upper-half complex plane there exists a counterpart theory.