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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 !

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  • $\begingroup$ A word of warning: you first need to think about what you mean by multiplying elements of ${\mathbb R}^n$ when $n\geq 3$. $\endgroup$
    – Yemon Choi
    Commented May 29, 2012 at 0:19
  • $\begingroup$ @ Yemon Choi : I meant the following map in $\mathbb{R}^3 \prod \mathbb{R}^3 \to \mathbb{R}^3 $ : $P(v,w)= P (v=(r_1,\theta_1,\phi_1), w=(r_2,\theta_2, \phi_2))= (r_1.r_2, \theta_1+\theta_2, \phi_1+\phi_2 )$. This corresponds to the geometric definition of product of two complex numbers in $\mathbb{C}$. $\endgroup$
    – Analysis Now
    Commented Jun 7, 2012 at 3:41
  • $\begingroup$ @ Yemon Choi : ...complex numbers written in the polar co-ordinates $(r, \theta)$, with the identification : $z= re^{i \theta}$ $\endgroup$
    – Analysis Now
    Commented Jun 7, 2012 at 3:44

2 Answers 2

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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".

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  • $\begingroup$ @ Steven Gubkin : True, but that way you can define inner functions in even real dimensions only, how can you define something like that in $\mathbb{R}^n$, where $n$ is odd ? You don't have the complex structure there, but you can still talk about conformal maps, which are merely angle-preserving maps. And if you have a suitable definition of product in $\mathbb{R}^n, n$ even or odd, then you can use that product to define some product of conformal maps. $\endgroup$
    – Analysis Now
    Commented Jun 7, 2012 at 3:48
  • $\begingroup$ @ Steven Gubkin : Also, Blaschke products in $\mathbb{C}$ are just proper holomorphic functions on the open unit disk in $\mathbb{C}$. But I am aware of a ( which I guess non-trivial ) result that states : proper holomorphic functions from the open unit disk in $\mathbb{C}^n, n > 1$ to itself are biholomorphism of the open unit disk. $\endgroup$
    – Analysis Now
    Commented Jun 7, 2012 at 3:53
  • $\begingroup$ ** By the functions on the open unit disk I meant the functions from the open unit disk to itself . $\endgroup$
    – Analysis Now
    Commented Jun 7, 2012 at 3:54
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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.

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