Hello, as we know, the (finite) Blaschke product $P$ in $\mathbb{C}$ or in $ \mathbb{R}^2 $ is defined by $\prod_{j=1}^{k} \frac{z-a_j}{1-\bar{a_j}z}, a_j \in \mathbb{D}$. I was wondering whether there are similar definitions of (finite) Blaschke products in higher ( real ) dimensions,in $\mathbb{R}^n$, $n \geq 3$.

I think, to construct Blaschke product in higher dimensions, we need to keep in mind that $P$ maps $\mathbb{B}^n$ to itself, and $|P(x)|\to 1$ as $ |x| \to 1 $ and $P(\frac{1}{\bar{z}})= \frac{1}{\bar{P(z)}}$.

I myself was trying to define a product of two vectors in $\mathbb{R}^3$ by defining the map using the spherical polar co-ordinates : $P(v,w)= P (v=(r_1,\theta_1,\phi_1), w=(r_2,\theta_2, \phi_2))= (r_1.r_2, \theta_1+\theta_2, \phi_1+\phi_2 )$, resembling the multiplication of complex numbers. But then the question becomes, in order to define Blachke product of say at least two maps, what kind of maps we should really multiply. There is no concept of holomorphic maps on $\mathbb{R}^3$, but we can try to replace them by conformal automorphism of $\mathbb{B}^n$, keeping in mind that $ \frac{z-a_j}{1-\bar{a_j}z}, a_j \in \mathbb{D}$ are conformal automorphisms of $ \mathbb{D}= \mathbb{B}^2$.

Before proceeding more, I was checking with the math community whether this is the standard way to define higher dimensional Blaschke products, or there are other standard way(s) to define them.Please let me know or cite any reference(s) you know. Thanks !