In mathematics, the Blaschke product in complex analysis is an analytic function designed to have zeroes at an infinite sequence of given complex numbers

a₀, a₁, ...

inside the unit disc. Given such a sequence, subject to the condition that

Σ (1 − |a_n|)

is convergent, define

B(z) = Π B(a_n, z)

where the factor

provided a ≠ 0. Here a^* is the complex conjugate of a. When a = 0 take B(0,z) = z.

Then the Blaschke product B(z) is analytic in the open unit, and is zero at the a_n only (with multiplicity counted). It is named for Wilhelm Blaschke, who described it in a paper in 1915. This seemingly peculiar function takes on importance because of its relationship to the study of Hardy spaces.

The sequence of a_n satisfying the convergence criterion above is sometimes called a Blaschke sequence.

See also

• Hardy space

References

• Peter Colwell, Blaschke Products - Bounded Analytic Functions (1985), University of Michigan Press, Ann Arbor. ISBN 0-472-10065-3