The Blaschke product is a class of analytic functions defined on the open unit disc $ \mathbb{D} = {z \in \mathbb{C}: |z|<1} $ that are constructed as (possibly infinite) products of elementary Möbius transformations mapping the disc onto itself.
Definition
For a sequence ${a_n}$ of points in $\mathbb{D}$ satisfying the Blaschke condition
$$ \sum_{n=1}^{\infty} (1-|a_n|) < \infty , $$
the (infinite) Blaschke product associated with ${a_n}$ is
$$ B(z)=\prod_{n=1}^{\infty} b_{a_n}(z),\qquad b_{a}(z)=\frac{|a|}{a},\frac{a-z}{1-\overline{a},z} \quad (a eq0),\qquad b_{0}(z)=z . $$
If the sequence is finite, the product terminates after a finite number of factors and is called a finite Blaschke product.
Basic properties
- Each factor $b_{a}(z)$ is a Möbius transformation that maps $\mathbb{D}$ onto itself and has a simple zero at $z=a$.
- The product $B(z)$ is analytic on $\mathbb{D}$ and extends to a function of modulus one almost everywhere on the unit circle $\partial\mathbb{D}$.
- For a finite Blaschke product of degree $n$, $B$ is a rational function of degree $n$ that is an inner function: $|B(e^{i\theta})|=1$ for all $\theta$ where the boundary values exist.
- The zeros of $B$ in $\mathbb{D}$ are precisely the points ${a_n}$, counted with multiplicity.
Historical context
The concept is named after the German mathematician Wilhelm Blaschke (1885–1962), who studied these products in the early 20th century in connection with the theory of Hardy spaces and conformal mapping.
Related concepts
- Inner functions: Functions analytic in $\mathbb{D}$ whose radial boundary values have modulus one almost everywhere; Blaschke products form a principal subclass.
- Hardy spaces $H^p$: Blaschke products appear in the canonical factorization of functions in $H^p$.
- Singular inner functions: Complementary inner factors defined via singular measures on $\partial\mathbb{D}$.
Applications
- Function theory: Provide explicit examples of inner functions and are used in factorization theorems (e.g., the inner‑outer factorization of Hardy‑space functions).
- Operator theory: Serve as characteristic functions of certain contraction operators (via the Sz.-Nagy–Foiaș model).
- Signal processing: Finite Blaschke products are employed in filter design because of their all‑pass or phase‑preserving properties.
References
- W. Rudin, Real and Complex Analysis, 3rd ed., McGraw‑Hill, 1987.
- P. D. Lax, Functional Analysis, Wiley, 2002 – Section on Hardy spaces and inner functions.
- J. B. Garnett, Bounded Analytic Functions, Graduate Texts in Mathematics 236, Springer, 2007.
