( finite ) Blaschke product in higher dimensions ? 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 !
 A: A key word to look for is "inner function".  These are bounded analytic functions on the ball where the limit along any radius to the boundary exists with modulus equal to one almost everywhere. These are the useful equivalent of Blashke products to $\mathbb{C}^n$ - they are useful in rigging up holomorphic functions with particular boundary conditions.  The construction of such functions was quite a feat - they were thought not to exist for a long time. It was conjectured that they didn't exist in 1965, and the first examples were constructed in 1982.
A good reference is chapter 9 of Krantz's "Several Complex Variables".  
A: There is a natural generalization of finite Blaschke products in
higher complex dimensions.  Consider
automorphisms (a-L_a z)/(1 - ) of the unit ball.
One takes their tensor product.
For example, the generalization of z-> z^m in the unit disk
is the map z-> z^{\otimes m} from the unit ball to the unit ball
in a larger dimension.
The result, for any tensor product of automorphisms, is a proper holomorphic map to a much higher dimensional ball.
See various papers by J. P. D'Angelo and by D'Angelo and J. Lebl.
See also Chapter 5 of "Several Complex Variables and the Geometry of Real Hypersurfaces" by John P. D'Angelo.
