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 $$f=e_a⟨f,e_a⟩+B_a\ast {\mathcal{S}}_{a}f,$$ 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.