Blaschke product

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A sequence of points ${\displaystyle (a_{n})}$ inside the unit disk is said to satisfy the Blaschke condition when

${\displaystyle \sum _{n}(1-|a_{n}|)<\infty .}$

Given a sequence obeying the Blaschke condition, the Blaschke product is defined as

${\displaystyle B(z)=\prod _{n}B(a_{n},z)}$

with factors

${\displaystyle B(a,z)={\frac {|a|}{a}}{\frac {a-z}{1-{\overline {a}}z}}}$

provided ${\displaystyle a\neq 0}$. Here ${\displaystyle {\overline {a}}}$ is the complex conjugate of ${\displaystyle a}$. When ${\displaystyle a=0}$, take ${\displaystyle B(0,z)=z}$.

The Blaschke product ${\displaystyle B(z)}$ defines a function analytic in the open unit disc and vanishing exactly at the points ${\displaystyle a_{n}}$, with multiplicity counted. Furthermore, it belongs to the Hardy class ${\displaystyle H^{\infty }}$.^[3]

The sequence ${\displaystyle (a_{n})}$ satisfying the convergence criterion above is sometimes called a Blaschke sequence.

A theorem of Gábor Szegő states that if ${\displaystyle f\in H^{1}}$, the Hardy space with integrable norm, and if ${\displaystyle f}$ is not identically zero, then the zeroes of ${\displaystyle f}$, which are at most countable, satisfy the Blaschke condition.

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Finite Blaschke products can be characterized as analytic functions on the unit disc in the following way.^[4] Assume that ${\displaystyle f}$ is analytic on the open unit disc and can be extended to a continuous function on the closed unit disc

${\displaystyle {\overline {\Delta }}=\{z\in \mathbb {C}$ :|z|\leq 1\}}

that maps the unit circle to itself. Then ${\displaystyle f}$ is equal to a finite Blaschke product

${\displaystyle B(z)=\zeta \prod _{i=1}^{n}\left({\frac {z-a_{i}}{1-{\overline {a_{i}}}z}}\right)^{m_{i}},}$

where ${\displaystyle \zeta }$ lies on the unit circle, ${\displaystyle |a_{i}|<1}$, and ${\displaystyle m_{i}}$ is the multiplicity of the zero ${\displaystyle a_{i}}$. In particular, if ${\displaystyle f}$ satisfies the condition above and has no zeros inside the unit circle, then ${\displaystyle f}$ is constant. This fact is also a consequence of the maximum principle for harmonic functions, applied to the harmonic function ${\displaystyle \log |f(z)|}$.

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For ${\displaystyle 0<\rho <1}$, consider the Blaschke factor

${\displaystyle b_{\rho }(z)={\frac {z-\rho }{1-\rho z}}}$

and write

${\displaystyle b_{\rho }(z)^{n}=\sum _{k\geq 0}{\widehat {b_{\rho }^{n}}}(k)z^{k}.}$

Set

${\displaystyle \alpha _{0}={\frac {1-\rho }{1+\rho }}.}$

When ${\displaystyle n}$ is large and ${\displaystyle k}$ is proportional to ${\displaystyle n}$, the asymptotic behavior of these coefficients changes at the two indices ${\displaystyle \alpha _{0}n}$ and ${\displaystyle \alpha _{0}^{-1}n}$. If ${\displaystyle k/n}$ remains in a compact subinterval of ${\displaystyle (\alpha _{0},\alpha _{0}^{-1})}$, the coefficients have oscillatory stationary-phase asymptotics of order ${\displaystyle n^{-1/2}}$. Outside this interval they decay exponentially. An Airy-type asymptotic near the transition point ${\displaystyle \alpha _{0}^{-1}n}$, approached from inside the interval ${\displaystyle (\alpha _{0}n,\alpha _{0}^{-1}n)}$, is established in this setting.^[5] More generally, near either endpoint, on the scale ${\displaystyle |k-\alpha _{0}^{\pm 1}n|=O(n^{1/3})}$, the transition is described by the Airy function, and the largest coefficients are of order ${\displaystyle n^{-1/3}}$.^[6]

For ${\displaystyle 1\leq p\leq \infty }$, the corresponding asymptotic orders of the ${\displaystyle \ell ^{p}}$-norms of the Taylor coefficients are

${\displaystyle \left\|{\widehat {b_{\rho }^{n}}}\right\|_{\ell ^{p}}\asymp {\begin{cases}n^{\,1/p-1/2},&1\leq p<4,\\[4pt]\left({\dfrac {\log n}{n}}\right)^{1/4},&p=4,\\[8pt]n^{\,1/(3p)-1/3},&4<p\leq \infty ,\end{cases}}}$

where ${\displaystyle 1/\infty =0}$ and the implicit constants depend on ${\displaystyle p}$ and ${\displaystyle \rho }$.^[5]

For general finite Blaschke products, the asymptotic size of the largest coefficient of ${\displaystyle B^{n}}$ is similarly determined by the maximal degeneracy of the stationary points of the boundary phase of ${\displaystyle B}$.^[7]

• Hardy space
• Weierstrass product
• Complex analysis