Adaptive Fourier Decomposition of Slice Regular Functions

Abstract

In the slice Hardy space over the unit ball of quaternions, we introduce the slice hyperbolic backward shift operator \(\mathcal S_a\) with the decomposition process $$\begin{aligned} f=e_a\langle f, e_a\rangle +B_{a}*\mathcal S_a f, \end{aligned}$$where \(e_a\) denotes the slice normalized Szegö kernel and \( B_a \) the slice Blaschke factor. Iterating the above decomposition process, a corresponding maximal selection principle gives rise to the slice adaptive Fourier decomposition. This leads to a adaptive slice Takenaka–Malmquist orthonormal system.