Adaptive Fourier decomposition of slice regular functions

TL;DR

This paper introduces an adaptive Fourier decomposition method for slice regular functions in quaternionic analysis, utilizing a new backward shift operator and orthonormal systems to improve function representation.

Contribution

It develops a novel adaptive Fourier decomposition framework for slice regular functions using a slice hyperbolic backward shift operator and maximal selection principles.

Findings

Established a new decomposition process involving the slice hyperbolic backward shift operator.

Constructed an adaptive slice Takenaka-Malmquist orthonormal system.

Provided a maximal selection principle for the decomposition process.

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 $$f=e_a\langle f, e_a\rangle+B_{a}*\mathcal S_a f,$$ where $e_a$ denotes the slice normalized Szeg\"o 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.