We know that the set of functions $\{1,\cos x, \sin x, \cos 2x, \sin 2x, ... \; | \,x \in \mathbb{R} \}$ is a basis in the space $L^2_\mathbb{R}[-\pi,\pi]$ .
Given a quaternion $z \in \mathbb{H}$ we can analogously define the functions $$ \cos nz= \sum_{k=0}^\infty\dfrac{(-1)^k\,(nz)^{2k}}{(2k)!} = \dfrac{e^{nz}+e^{-nz}}{2} $$ $$ \sin nz = \sum_{k=0}^\infty\dfrac{(-1)^k\,(nz)^{2k+1}}{(2k+1)!}= \dfrac{e^{nz}-e^{-nz}}{2\mathbf{i}} $$ where $\mathbf{i}$ is one of the immaginary unit of $\mathbb{H}$.
Is the set $\{1,\cos z, \sin z, \cos 2z, \sin 2z, ... \; | \,z \in \mathbb{H} \}$ a basis for some suitable space of functions $f:\mathbb{H}\rightarrow \mathbb{H}$ ?