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Blaschke ProductIn 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 − |an|)
is convergent, define - B(z) = Π B(an, 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 an 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 an satisfying the convergence criterion above is sometimes called a Blaschke sequence. See also References - Peter Colwell, Blaschke Products - Bounded Analytic Functions (1985), University of Michigan Press, Ann Arbor. ISBN 0-472-10065-3
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