N-best adaptive Fourier decomposition for slice hyperholomorphic functions

Abstract

The purpose of this article is to establish the N-best adaptive Fourier decomposition for slice hyperholomorphic functions in the slice hyperholomorphic quaternionic Hardy space $H^2\left(H^{+}\right)$ and $H^2\left(B\right)$, where $H^{+}$ is the right half space and $B$ is the Euclidean unit ball of quaternions. We prove the existence of the N-best approximation problem for $H^2$ functions which requires considering multiple parameters of the slice Takenaka-Malmquist system simultaneously. In the non-commutative quaternion field our proof relies on the limit behavior for the slice Takenaka-Malmquist system which is obtained through separating quaternionic Blaschke factors from elements of the system, that is quite different to the complex variable and several complex variables cases. Technically, it is more subtle for the right half space $H^{+}$.