a Blaschke product
/ BLAHSH-kuh /
Weierstrass products build entire functions with prescribed zeros on the whole plane, but they grow freely and have no built-in size limit. On the unit disk there is a more delicate construction tailored to BOUNDED holomorphic functions: the Blaschke product. It manufactures a holomorphic function on the disk with prescribed zeros that nevertheless stays bounded by 1 in modulus — a feat impossible for a Weierstrass-style product, which would blow up.
The building block is the Blaschke factor for a zero at a point a inside the disk: B_a(z) = (|a|/a) times (a - z)/(1 - (a-bar) z). This Mobius-type factor vanishes exactly at z = a, has modulus exactly 1 on the boundary circle |z| = 1, and the unimodular constant |a|/a is a phase chosen so that B_a(0) > 0. A Blaschke product is a product of such factors, one per prescribed zero a_n (with a factor z^m thrown in for a zero of order m at the origin): B(z) = z^m times the product over n of (|a_n|/a_n)(a_n - z)/(1 - (a_n-bar) z). Each factor is bounded by 1 on the disk, so the product, when it converges, is a bounded holomorphic function with exactly the zeros a_n.
Blaschke products are the disk's analogue of the polynomial-or-Weierstrass factor, and they are the canonical zero-divisors of bounded analytic functions. The central theorem (factorization in the Hardy space and Nevanlinna class) says any bounded holomorphic function on the disk factors as a Blaschke product carrying its zeros, times a non-vanishing piece. They appear all over: in describing disk automorphisms (a single Blaschke factor IS a disk automorphism), in interpolation problems, and in operator theory. The catch is that, unlike Weierstrass products, a Blaschke product converges only under a definite condition on the zeros — the Blaschke condition.
A single Blaschke factor B_a(z) = (a - z)/(1 - (a-bar) z) (taking the phase to be 1) with a = 1/2 is z |-> (1/2 - z)/(1 - z/2). It vanishes at z = 1/2, maps the unit disk onto itself bijectively, and has |B_a(z)| = 1 exactly on |z| = 1. This is simultaneously a one-zero Blaschke product and an automorphism of the disk.
One Blaschke factor: vanishes at a inside the disk, unimodular on the boundary, a disk automorphism.
A finite Blaschke product always converges and is bounded by 1; an infinite one converges to a nonzero bounded function only if the zeros satisfy the Blaschke condition. If they do not, the product collapses to the constant 0 — there is no bounded holomorphic function with those zeros.